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True or False If a function \(f\) is continuous on the interval \([ a,b]\), differentiable on the interval \(( a,b)\), and changes from an increasing function to a decreasing function at the point \((c, f( c))\), then \((c,f( c))\) is an inflection point of \(f\).

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True or False Suppose \(c\) is a critical number of \(f\) and \((a,b)\) is an open interval containing \(c\). If \(f^\prime (x)\) is positive on both sides of \(c\), then \(f( c)\) is a local maximum value.

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Multiple Choice Suppose a function \(f\) is continuous on a closed interval \([a,b]\) and differentiable on the open interval \((a,b)\). If the graph of \(f\) lies above each of its tangent lines on the interval \(( a,b)\), then \(f\) is [\(({\bf a})\) concave up, \(({\bf b})\) concave down, \(({\bf c})\) neither] on \(( a,b)\).

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Multiple Choice If the acceleration of an object in rectilinear motion is negative, then the velocity of the object is [\(({\bf a})\) increasing, \(({\bf b})\) decreasing, \(({\bf c})\) neither].

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Multiple Choice Suppose \(f\) is a function that is differentiable on an open interval containing \(c\) and the concavity of \(f\) changes at the point \((c,f( c))\). Then \((c,f( c))\) is a(n) [\(({\bf a})\) inflection point, \(({\bf b})\) critical point, \(({\bf c})\) both \(({\bf d})\) neither] of \(f\).

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Multiple Choice Suppose a function \(f\) is continuous on a closed interval \([a,b]\) and both \(f^\prime\) and \(f^{\prime \prime}\) exist on the open interval \((a,b)\). If \(f^{\prime \prime} (x) >0\) on the interval \(( a,b)\), then \(f\) is [\(({\bf a})\) increasing, \(({\bf b})\) decreasing, \(({\bf c})\) concave up, \(({\bf d})\) concave down] on \(( a,b)\).

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True or False Suppose \(f\) is a function for which \(f^\prime\) and \(f^{\prime \prime}\) exist on an open interval \((a,b)\) and suppose \(c,\) \(a<c<b,\) is a critical number of \(f\). If \(f^{\prime \prime} ( c) =0\), then the Second Derivative Test cannot be used to determine if there is a local extremum at \(c\).

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True or False Suppose a function \(f\) is differentiable on the open interval \((a,b)\). If either \(f^{\prime \prime} (c) =0\) or \(f^{\prime \prime}\) does not exist at the number \(c\) in \(( a,b) ,\) then \((c, f( c))\) is an inflection point of \(f.\)

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\(f(x)=x^{3}-6x^{2}+2\)

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\(f(x)=x^{3}+6x^{2}+12x+1\)

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\(f(x)=3x^{4}-4x^{3}\)

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\(h(x)=x^{4}+2x^{3}-3\)

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\(f(x)= ( 5-2x ) e^{x}\)

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\(f(x)=(x-8) e^{x}\)

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\(g(u)=u^{-1}e^{u}\)

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\(f(x)=x^{-2}e^{x}\)

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\(f(x)=x^{2/3}+x^{1/3}\)

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\(f(x)=\dfrac{1}{2}x^{2/3}-x^{1/3}\)

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\(g(x)=x^{2/3}(x^{2}-4)\)

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\(f(x)=x^{1/3}(x^{2}-9)\)

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\(f(x)=\dfrac{\ln x}{x^{3}}\)

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\(h(x)=\dfrac{\ln x}{ \sqrt{x^{3}}}\)

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\(g(x)=|{x^{2}-1}|\)

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\(f(x)=|{x^{2}-4}|\)

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\(f(\theta )=\sin \theta -2\cos \theta\)

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\(f(x)=x+2\sin x\)

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\(s=t^{2}-2t+3\)

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\(s=2t^{2}+8t-7\)

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\(s=2t^{3}+6t^{2}-18t+1\)

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\(s=3t^{4}-16t^{3}+24t^{2}\)

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\(s=2t-\dfrac{6}{t}, \quad t>0\)

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\(s=3 \sqrt{t}-\dfrac{1}{ \sqrt{t}},\quad t>0\)

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\(s=2\, \sin ( 3t) ,\quad 0\leq t\leq \dfrac{2\pi }{3}\)

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\(s=3\, \cos ( \pi t) ,\quad 0\leq t\leq 2\)

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\(f(x)=x^{2}-2x+5\)

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\(f(x)=x^{2}+4x-2\)

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\(f(x)=x^{3}-9x^{2}+2\)

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\(f(x)=x^{3}-6x^{2}+9x+1\)

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\(f(x)=x^{4}-4x^{3}+10\)

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\(f(x)=3x^{4}-8x^{3}+6x+1\)

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\(f(x)= x^{2/3}e^{x}\)

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\(f(x)= x^{2/3}e^{-x}\)

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\(f(x)=\dfrac{\ln x}{x^{3}}\)

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\(f(x)=\dfrac{\ln x}{ \sqrt{x^{3}}}\)

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\(f(x)=x+\dfrac{1}{x}\)

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\(f(x)=2x^{2}-\dfrac{1}{x}\)

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\(f(x)=3x^{1/3}+2x\)

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\(f(x)=x^{4/3}-8x^{1/3}\)

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\(f(x)=3-\dfrac{4}{x}+\dfrac{4}{x^{2}}\)

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\(f(x)=(x-1)^{3/2}\)

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\(f(x)=2x^{3}-6x^{2}+6x-3\)

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\(f(x)=2x^{3}+9x^{2}+12x-4\)

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\(f(x)=x^{4}-4x\)

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\(f(x)=x^{4}+4x\)

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\(f(x)=5x^{4}-x^{5}\)

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\(f(x)=4x^{6}+6x^{4}\)

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\(f(x)=3x^{5}-20x^{3}\)

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\(f(x)=3x^{5}+5x^{3}\)

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\(f(x)=x^{2}e^{x}\)

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\(f(x)=x^{3}e^{x}\)

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\(f(x)=\dfrac{e^{x}+e^{-x}}{2}\)

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\(f(x)=\dfrac{e^{x}-e^{-x}}{2}\)

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\(f(x)=6x^{4/3}-3x^{1/3}\)

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\(f(x)=x^{2/3}-x^{1/3}\)

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\(f(x)=x^{2/3}(x^{2}-8)\)

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\(f(x)=x^{1/3}(x^{2}-2)\)

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\(f(x)=x^{2}-\ln x\)

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\(f(x)=\ln x-x\)

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\(f(x)=\dfrac{x}{({1+x^{2}})^{5/2}}\)

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\(f(x)=\dfrac{ \sqrt{x}}{1+x}\)

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\(f(x)=x^{2} \sqrt{1-x^{2}}\)

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\(f(x)=x \sqrt{1-x}\)

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\(f(x)={\sin ^{2}} x\)

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\(f(x)=\cos ^{2}x\)

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\(f(x)=x-2\sin x, \quad 0\leq x\leq 2\pi\)

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\(f(x)=2\, \cos ^{2}x-\!\sin ^{2}x, \quad 0\leq x\leq 2\pi\)

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\(f(x)=-2x^{3}+15x^{2}-36x+7\)

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\(f(x)=x^{3}+10x^{2}+25x-25\)

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\(f(x)=(x-3) ^{2}e^{x}\)

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\(f(x)=(x+1) ^{2}e^{-x}\)

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\(f\) is concave up on \((-\infty ,\infty )\), increasing on \((-\infty ,0)\), decreasing on \(( 0,\infty )\), and \(f(0)=1\).

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\(f\) is concave up on \((-\infty ,0)\), concave down on \(( 0,\infty )\), decreasing on \((-\infty ,0)\), increasing on \(( 0,\infty )\), and \(f(0)=1\).

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\(f\) is concave down on \((-\infty ,1)\), concave up on \((1,\infty )\), decreasing on \((-\infty ,0)\), increasing on \(( 0,\infty )\), \(f(0)=1\), and \(f(1)=2\).

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\(f\) is concave down on \((-\infty ,0)\), concave up on \(( 0,\infty )\), increasing on \((-\infty ,\infty )\), and \(f(0)=1\) and \(f(1)=2\).

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\({{f^\prime }(x)>0}\) if \({x<0}\); \({{f^\prime }(x)<0}\) if \({x>0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x<0}\); \({f^{\prime \prime} (x)>0}\) if \({x>0}\) and \({f(0)=1}\).

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\({{f^\prime }(x)>0}\) if \({x<0}\); \({{f^\prime }(x)<0}\) if \({x>0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x<0}\); \({{f^{\prime \prime} }(x)<0}\) if \({x>0}\) and \({f(0)=1}\).

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\({{f^{\prime \prime} }(0)=0}\); \({{f^\prime }(0)=0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x<0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x>0}\) and \({f(0)=1}\).

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\({{f^{\prime \prime} }(0)=0}\); \({{f^\prime }(x)>0}\) if \({x\neq 0}\); \({{f^{\prime \prime} }(x)<0}\) if \({x<0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x>0}\) and \({f(0)=1}\).

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\({{f^\prime }(0)=0}\); \({f^\prime }(x) <{0}\) if \({x\neq 0}\); \({{f^{\prime \prime} }(x)>0}\) if \({x<0}\); \({{f^{\prime \prime} }(x)<0}\) if \({x>0}\); \({f(0)=1}\).

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\({{f^{\prime \prime} }(0)=0}\); \({{f^\prime }(0)=\dfrac{1}{2}}\); \({{f^{\prime \prime} }(x)>0}\) if \({x<0}\); \(\ {{f^{\prime \prime} }(x)<0}\) if \({x>0}\) and \({f(0)=1}\).

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\(f^\prime ( 0)\) does not exist; \(f^{\prime \prime} (x) >0\) if \(x<0\); \(f^{\prime \prime} (x)>0\) if \(x>0\) and \(f(0)=1\).

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\(f^\prime ( 0)\) does not exist; \(f^{\prime \prime} (x) <0\) if \(x<0\); \(f^{\prime \prime} (x)>0\) if \(x>0\) and \(f(0)=1\).

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\(f(x) =e^{-( x-2) ^{2}}\)

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\(f(x) =x^{2} \sqrt{5-x}\)

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\(f(x) =\dfrac{2-x}{2x^{2}-2x+1}\)

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\(f(x) =\dfrac{3x}{x^{2}+3x+5}\)

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Critical Number Show that \(0\) is the only critical number of \(f(x)= \sqrt[3]{x}\) and that \(f\) has no local extrema.

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Critical Number Show that \(0\) is the only critical number of \(f(x)= \sqrt[3]{x^{2}}\) and that \(f\) has a local minimum at \(0\).

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Inflection Point For the function \(f(x)=ax^{3}+bx^{2}\), find \(a\) and \(b\) so that the point \((1,6)\) is an inflection point of \(f\).

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Inflection Point For the cubic polynomial function \(f(x)=ax^{3}+bx^{2}+cx+d\), find \(a\), \(b\), \(c\), and \(d\) so that \(0\) is a critical number, \(f( 0) =4\), and the point \((1,-2)\) is an inflection point of \(f\).

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Public Health In a town of \(50{,}000\) people, the number of people at time \(t\) who have influenza is \(N(t) =\dfrac{10,000}{1+9999e^{-t}}\), where \(t\) is measured in days. Note that the flu is spread by the one person who has it at \(t=0\).

1. Find the rate of change of the number of infected people.
2. When is \(N' (t)\) increasing? When is it decreasing?
3. When is the rate of change of the number of infected people a maximum?
4. Find any inflection points of \(N(t)\).
5. Interpret the result found in (d).

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Business The profit \(P\) (in millions of dollars) generated by introducing a new technology is expected to follow the logistic function \(P(t) =\dfrac{300}{1+50e^{-0.2t}}\), where \(t\) is the number of years after its release.

1. When is annual profit increasing? When is it decreasing?
2. Find the rate of change in profit.
3. When is the rate of change in profit \(P'\) increasing? When is it decreasing?
4. When is the rate of change in profit a maximum?
5. Find any inflection points of \(P(t)\).
6. Interpret the result found in (e).

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Population Model The following data represent the population of the United States:

An ecologist finds the data fit the logistic function \[ P(t) = \dfrac{762{,}176{,}717.8}{1 + 8.7427 e^{-0.0162t}}. \]

1. Draw a scatterplot of the data using the years since \(1900\) as the independent variable and population as the dependent variable.
2. Verify that \(P\) is the logistic function of best fit.
3. Find the rate of change in population.
4. When is \(P' (t)\) increasing? When is it decreasing?
5. When is the rate of change in population a maximum?
6. Find any inflection points of \(P(t)\).
7. Interpret the result found in (f).

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Biology The amount of yeast biomass in a culture after \(t\) hours is given in the table below.

Source: Tor Carlson, Uber Geschwindigkeit und Grosse der Hefevermehrung in Wurrze, Biochemische Zeitschrift, Bd. 57, 1913, pp. 313–334.

The logistic function \(y=\dfrac{663.0}{1+71.6e^{-0.5470t}}\), where \(t\) is time, models the data.

1. Draw a scatterplot of the data using time \(t\) as the independent variable.
2. Verify that \(y\) is the logistic function of best fit.
3. Find the rate of change in biomass.
4. When is \(y^\prime (t)\) increasing? When is it decreasing?
5. When is the rate of change in the biomass a maximum?
6. Find any inflection points of \(y(t)\).
7. Interpret the result found in (f).

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U.S. Budget The United States budget documents the amount of money (revenue) the federal government takes in (through taxes, for example) and the amount (expenses) it pays out (for social programs, defense, etc.). When revenue exceeds expenses, we say there is a budget surplus, and when expenses exceed revenue, there is a budget deficit. The function \[ B=B(t) =-12.8t^{3}+163.4t^{2}-614.0t+390.6 \]

where \(0\leq t\leq 9\) approximates revenue minus expenses for the years 2000 to 2009, with \(t=0\) representing the year 2000 and \(B\) in billions of dollars.

1. Find all the local extrema of \(B\). (Round the answers to two decimal places.)
2. Do the local extreme values represent a budget surplus or a budget deficit?
3. Find the intervals on which \(B\) is concave up or concave down. Identify any inflection points of \(B\).
4. What does the concavity on either side of the inflection point(s) indicate about the rate of change of the budget? Is it increasing at an increasing rate? Increasing at a decreasing rate?
5. Graph the function \(B\). Given that the total amount the government paid out in 2011 was \(3.6\) trillion dollars, does \(B\) seem to be an accurate predictor for the budget for the years 2010 and beyond?

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If \(f(x)=ax^{3}+bx^{2}+cx+d\), \(a\neq 0\), how does the quantity \(b^{2}-3ac\) determine the number of potential local extrema?

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If \(f(x)=ax^{3}+bx^{2}+cx+d\), \(a\neq 0\), find \(a\), \(b\), \(c\), and \(d\) so that \(f\) has a local minimum at \(0\), a local maximum at \(4\), and the graph contains the points \((0,5)\) and \((4,33)\).

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Find the local minimum of the function \[ f(x)=\dfrac{2}{x}+\dfrac{8}{1-x},\quad 0<x<1. \]

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Find the local extrema and the inflection points of \(y= \sqrt{3}\sin x+\cos x\), \(0\leq x\leq 2\pi\).

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If \(x>0\) and \(n>1\), can the expression \(x^{n}-n(x-1)-1\) ever be negative?

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Why must the First Derivative Test be used to find the local extreme values of the function \(f(x) =x^{2/3}\)?

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Put It Together Which of the following is true of the function \[ f(x)=x^{2}+e^{-2x} \]

1. \(f\) is increasing, \({\bf (b)}\) \(f\) is decreasing, \({\bf (c)}\) \(f\) is discontinuous at \(0\)
2. \(f\) is concave up, \({\bf (e)}\) \(f\) is concave down

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Put It Together If a function \(f\) is continuous for all \(x\) and if \(f\) has a local maximum at \((-1,4)\) and a local minimum at \((3,-2)\), which of the following statements must be true?

1. The graph of \(f\) has an inflection point somewhere between \(x=-1\) and \(x=3\).
2. \(f^\prime (-1)=0\).
3. The graph of \(f\) has a horizontal asymptote.
4. The graph of \(f\) has a horizontal tangent line at \(x=3\).
5. The graph of \(f\) intersects both axes.

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Vertex of a Parabola If \(f(x)\,=\,ax^{2}\,+\,bx\,+\,c\), \(a\,\neq \,0\), prove that \(f\) has a local maximum at \(-\dfrac{b}{2a}\) if \(a<0\) and has a local minimum at \(-\dfrac{b}{2a}\) if \(a\,>\,0\).

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Show that \(\sin x\leq x\), \(\ 0\leq x\leq 2\pi\). (Hint: Let \(f(x)=x- \sin x\).)

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Show that \(1-\dfrac{x^{2}}{2}\leq \cos x\), \(\ 0\leq x\leq 2\pi\). (Hint: Use the result of Problem 119.)

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Show that \(2 \sqrt{x}>3-\dfrac{1}{x}\), for \(x>1\).

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Use calculus to show that \(x^{2}-8x+21>0\) for all \(x\).

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Use calculus to show that \(3x^{4}-4x^{3}-12x^{2}+40>0\) for all \(x\).

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Show that the function \(f(x)=ax^{2}+bx+c\), \(a\neq 0\), has no inflection points. For what values of \(a\) is \(f\) concave up? For what values of \(a\) is \(f\) concave down?

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Show that every polynomial function of degree 3 \(f(x)=ax^{3}+bx^{2}+cx+\) \(d\) has exactly one inflection point.

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Prove that a polynomial of degree \(n\geq\) \(3\) has at most \(( n-1)\) critical numbers and at most \(( n-2)\) inflection points.

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Show that the function \(f(x)=(x-a)^{n}, a\) a constant, has exactly one inflection point if \(n\geq 3\) is an odd integer.

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Show that the function \(f(x)=(x-a)^{n}, a\) a constant, has no inflection point if \(n\geq 2\) is an even integer.

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Show that the function \(f(x)=\dfrac{ax+b}{ax+d}\) has no critical points and no inflection points.

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First Derivative Test Proof Let \(f\) be a function that is continuous on some interval \(I\). Suppose \(c\) is a critical number of \(f\) and \((a,b)\) is some open interval in \(I\) containing \(c\). Prove that if \(f^\prime (x)<0\) for \(a<x<c\) and \(f^\prime (x) >0\) for \(c<x<b\), then \(f( c)\) is a local minimum value.

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First Derivative Test Proof Let \(f\) be a function that is continuous on some interval \(I\). Suppose \(c\) is a critical number of \(f\) and \((a,b)\) is some open interval in \(I\) containing \(c\). Prove that if \(f^\prime (x)\) has the same sign on both sides of \(c\), then \(f( c)\) is neither a local maximum value nor a local minimum value.

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Test of Concavity Proof Let \(f\) denote a function that is continuous on a closed interval \([a,b]\). Suppose \(f^\prime\) and \(f^{\prime \prime}\) exist on the open interval \((a,b)\). Prove that if \(f^{\prime \prime}(x)<0\) on the interval \((a,b)\), then \(f\) is concave down on \(( a,b)\).

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Second Derivative Test Proof Let \(f\) be a function for which \(f^\prime\) and \(f^{\prime \prime}\) exist on an open interval \((a,b)\). Suppose \(c\) lies in \(( a,b)\) and is a critical number of \(f\). Prove that if \(f^{\prime \prime} ( c) >0\), then \(f( c)\) is alocal minimum value.

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Find the inflection point of \(y=\left( x+1\right) \tan ^{-1}x\).

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Bernoulli’s Inequality Prove Bernoulli’s inequality: \(( 1+x) ^{n}>1+nx\) for \(x>-1\), \(x\neq\) \(0\), and \(n>1\). (Hint: Let \(f(x)=(1+x)^{n}-(1+nx)\).)