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\(f(x) =x^{4}-6x^{2}+10\)

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\(f(x) =x^{5}-3x^{3}+4\)

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

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\(f(x)=\dfrac{2}{x+2}\)

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

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

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

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

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

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

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

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

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\(f(x)=\dfrac{x^{4}+1}{x^{2}}\)

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

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\(xy=x^{2}+2\)

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

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

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

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

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

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\(f(x)=\sqrt{3-x}\)

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\(f(x)=x\sqrt{x+2}\)

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

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

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

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

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

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\(f(x)=\dfrac{1}{({x-1})({x+3})}\)

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

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\(f(x)=x^{5/3}-5x^{2/3}\)

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\(f(x)=\sin x-\cos x\)

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

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

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

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\(y=\ln x-x\)

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\(y=x\, \ln x\)

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

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\(y=\ln (x^{2}+2)\)

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

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

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

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\(f(x)=e^{1/x}\)

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

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\(f(x)=\dfrac{5-x}{x^{2}+3x+4}\)

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

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\(f(x)=x-\cos x\)

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

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

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\(f(x)=\sqrt[3]{\sin x}\)

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

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\(y^{2}=x^{2}( 6-x)\), \(y\geq 0\)

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\(y^{2}=x^{2}(4-x^{2}),\) \(y\geq 0\)

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\( \begin{array}[t]{l} f^\prime (2)\hbox{ does not exist}\\ f^\prime (3)=-1\\ f^{\prime \prime} (3)=0\\ f^\prime (5)=0 \\ f^{\prime \prime} (x)<0,\quad 2<x<3\\ f^{\prime \prime} (x)>0,\quad x>3 \end{array}\)

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\( \begin{array}[t]{l} f^\prime (2)=0 \\ f^{\prime \prime} (2)=0 \\ f^\prime (3)\hbox{ does not exist} \\ f^\prime (5)=0 \\ f^{\prime \prime} (x)>0,\quad 2<x<3 \\ f^{\prime \prime} (x)>0,\quad x>3 \end{array}\)

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\( \begin{array}[t]{l} f^\prime (2)=0 \\ \lim\limits_{x\rightarrow 3^{-}}\,f^\prime (x)=\infty \\ \lim\limits_{x\rightarrow 3^{+}}\,f^\prime (x)=\infty \\ f^\prime (5)=0 \\ f^{\prime \prime} (x)>0, \quad x<3 \\ f^{\prime \prime} (x)<0, \quad x>3 \end{array}\)

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\( \begin{array}[t]{l} f^\prime (2)=0 \\ \lim\limits_{x\rightarrow 3^{-}}\,f^\prime (x)=-\infty \\ \lim\limits_{x\rightarrow 3^{+}}{\,}f^\prime (x)=-\infty \\ f^\prime (5)=0 \\ f^{\prime \prime} (x)<0,\quad x<3 \\ f^{\prime \prime} (x)>0, \quad x>3 \end{array}\)

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Sketch the graph of a function \(f\) defined and continuous for \(-1\leq x\leq 2\) that satisfies the following conditions: \[ \begin{array}{@{\hspace*{-1.7pc}}l} f(-1)= 1 \quad f(1) = 2 \quad f(2) = 3 \quad f(0) = 0 \quad f \left( {\dfrac{1}{2}}\right) =3\\[3pt] \quad \lim\limits_{x\rightarrow -1^{+}}f^\prime (x) =-\infty\quad \lim\limits_{x\rightarrow 1^{-}} f^\prime (x) = -1 \quad \lim\limits_{x\rightarrow 1^{+}} f^\prime (x)=\infty\\ {f\hbox{ has a local minimum at }0} \quad {f\hbox{ has a local maximum at }\dfrac{1}{2}} \end{array} \]

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Graph of a Function Which of the following is true about the graph of \(y=\ln |x^{2}-1|\) in the interval \((-1,1)\)?

1. The graph is increasing.
2. The graph has a local minimum at \((0,0).\)
3. The graph has a range of all real numbers.
4. The graph is concave down.
5. The graph has an asymptote \(x=0.\)

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Properties of a Function Suppose \(f(x)=\dfrac{1}{x}+\ln x\) is defined only on the closed interval \(\dfrac{1}{e}\leq x\leq e\).

1. Determine the numbers \(x\) at which \(f\) has its absolute maximum and absolute minimum.
2. For what numbers \(x\) is the graph concave up?
3. Graph \(f\).

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Probability The function \(f(x)=\dfrac{1}{\sqrt{2\pi }} e^{-x^{2}/2}\), encountered in probability theory, is called the standard normal density function. Determine where this function is increasing and decreasing, find all local maxima and local minima, find all inflection points, and determine the intervals where \(f\) is concave up and concave down. Then graph the function.

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

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\(f(x)=x^{\sqrt{x}}\)

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\(f(x)=~x^{1/x}\)

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\(y=\dfrac{1}{x}\tan x\quad -\dfrac{\pi }{2}<x<\dfrac{\pi}{2}\)