Question 14.53

\(f(x,y) = x^2 - y^2 + xy\)

Question 14.54

\(f(x,y) = x^2 + y^2 - xy\)

Question 14.55

\(f(x,y) = x^2 + y^2 + 2xy\)

Question 14.56

\(f(x,y) = x^2 + y^2 + 3xy\)

Question 14.57

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

Question 14.58

\(f(x,y) = x^2 - 3xy + 5x - 2y + 6y^2 + 8\)

Question 14.59

\(f(x,y) = 3x^2 + 2xy + 2x + y^2 + y+4\)

Question 14.60

\(f(x,y) = \sin\,(x^2 + y^2)\) [consider only the critical point \((0,0)\)]

Question 14.61

\(f(x,y) = \cos\,(x^2 + y^2)\) [consider only the three critical points \((0,0), (\sqrt{\pi/2},\sqrt{\pi/2})\), and \((0,\sqrt{\pi})\)]

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Question 14.62

\(f(x,y) = y + x\sin y\)

Question 14.63

\(f(x,y) = e^x \cos y\)

Question 14.64

\(f(x,y) = (x-y)(xy-1)\)

Question 14.65

\(f(x,y) = xy + {\displaystyle\frac{1}{x}}+ {\displaystyle\frac{1}{y}}\)

Question 14.66

\(f(x,y) = \log\,(2+\sin xy)\)

Question 14.67

\(f(x,y) = x\sin y\)

Question 14.68

\(f(x,y) = (x+y)(xy+1)\)

Question 14.69

Find all local extrema for \(f(x, y)=8y^3+12x^2-24xy\).

Question 14.70

Let \(f(x, y, z)=x^2 +y^2 +z^2+kyz\).

• (a) Verify that (0, 0, 0) is a critical point for \(f\).
• (b) Find all values of \(k\) such that \(f\) has a local minimum at (0, 0, 0).

Question 14.71

Find and classify all critical points of \(f(x, y)= \frac{1}{3}x^3 +\frac{1}{3}y^3 - \frac{1}{2}x^2 -\frac{5}{2}y^2+6y+10\).

Question 14.72

Suppose (4, 2) is a critical point for the \(C^2\) function \(f(x, y)\). In each case, determine whether (4, 2) is a local maximum, a local minimum, or a saddle point.

• (a) \(f_{xx}(4, 2)= 1, \ f_{xy}(4, 2)=3, \ f_{yy}=5\)
• (b) \(f_{xx}(4, 2)= 2, \ f_{yx}(4, 2)=-1, \ f_{yy}=4\)
• (c) \(f_{xx}(4, 2)= -2, \ f_{xy}(4, 2)=1, \ f_{yy}=3\)

Question 14.73

Find the local maxima and minima for \(z=(x^2+3y^2)\,e^{1-x^2-y^2}\). (See Figure 2.1.15.)

Question 14.74

Let \(f(x,y)=x^2+y^2+kxy\). If you imagine the graph changing as \(k\) increases, at what values of \(k\) does the shape of the graph change qualitatively?

Question 14.75

An examination of the function \(f{:}\, \,{\mathbb R}^2 \to {\mathbb R}, (x,y) \mapsto (y-3x^2)(y-x^2)\) will give an idea of the difficulty of finding conditions that guarantee that a critical point is a relative extremum when Theorem 6 fails.footnote # Show that

• (a) the origin is a critical point of \(f\);
• (b) \(f\) has a relative minimum at \((0, 0)\) on every straight line through \((0, 0)\); that is, if \(g(t) = (at,bt)\), then \(f\circ g{:}\,\, {\mathbb R} \to {\mathbb R}\) has a relative minimum at 0, for every choice of \(a\) and \(b\);
• (c) the origin is not a relative minimum of \(f\).

Question 14.76

Let \(f(x,y)= Ax^2 + E\), where \(A\) and \(E\) are constants. What are the critical points of \(f\)? Are they local maxima or local minima?

Question 14.77

Let \(f(x,y) = x^2 - 2xy + y^2\). Here \(D=0\). Can you say whether the critical points are local minima, local maxima, or saddle points?

Question 14.78

Let \(f(x, y)=ax^2+bx^2\), where \(a, b \neq 0\).

• (a) Show that (0, 0) is the only critical point for \(f\).
• (b) Determine the nature of this critical point in terms of \(a\) and \(b\).

Question 14.79

Suppose \(f \colon \mathbb{R}^3 \to \mathbb{R}\) is \(C^2\), and that \(\textbf{x}_0\) is a critical point for \(f\). Suppose \({\it Hf}(\textbf{x}_0)(\textbf{h}) = h_1^2+h_2^2+h_3^2+4h_2h_3\). Does \(f\) have a local maximum, minimum, or saddle at \(\textbf{x}_0\)?

Question 14.80

Find the point on the plane \(2x-y+2z = 20\) nearest the origin.

Question 14.81

Show that a rectangular box of given volume has minimum surface area when the box is a cube.

Question 14.82

Show that the rectangular parallelepiped with fixed surface area and maximum volume is a cube.

Question 14.83

Write the number 120 as a sum of three numbers so that the sum of the products taken two at a time is a maximum.

Question 14.84

Show that if \((x_0,y_0)\) is a critical point of a quadratic function \(f(x,y)\) and \(D<0\), then there are points \((x,y)\) near \((x_0,y_0)\) at which \(f(x,y) > f(x_0,y_0)\) and, similarly, points for which \(f(x,y) < f(x_0,y_0)\).

Question 14.85

Let \(f(x, y)=x^6+x^2+y^6, \; g(x, y)=-x^6-x^2-y^6, \; h(x, y)=x^6-x^4+y^6\).

• (a) Show that (0, 0) is a degenerate critical point for \(f, g\), and \(h\).
• (b) Show that (0, 0) is a local minimum for \(f\), a local maximum for \(g\), and a saddle for \(h\).

Question 14.86

Let \(f(x, y)=5ye^x-e^{5x}-y^5\).

• (a) Show that \(f\) has a unique critical point and that this point is a local maximum for \(f\).
• (b) Show that \(f\) is unbounded on the \(y\) axis, and thus has no global maximum. [Note that for a function \(g(x)\) of a single variable, a unique critical point which is a local extremum is necessarily a global extremum. This example shows that this is not the case for functions of several variables.]

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Question 14.87

Determine the nature of the critical points of the function \[ f(x,y,z) = x^2 + y^2 + z^2 + xy. \]

Question 14.88

Let \(n\) be an integer greater than 2 and set \(f(x,y) = ax^n + cy^n\), where \(ac\neq 0\). Determine the nature of the critical points of \(f\).

Question 14.89

Determine the nature of the critical points of \(f(x,y) = x^3 + y^2 - 6xy + 6x + 3y\).

Question 14.90

Find the absolute maximum and minimum values of the function \(f(x,y) = (x^2 + y^2)^4\) on the disc \(x^2 + y^2 \le 1\). (You do not have to use calculus.)

Question 14.91

Repeat Exercise 38 for the function \(f(x,y) = x^2 + xy + y^2\).

Question 14.92

A curve \(C\) in space is defined implicitly on the cylinder \(x^2 + y^2 =1\) by the additional equation \(x^2 - xy + y^2 - z^2 = 1\). Find the point or points on \(C\) closest to the origin.

Question 14.93

Find the absolute maximum and minimum values for \(f(x,y) = \sin x + \cos y\) on the rectangle \(R\) defined by \(0 \le x \le 2\pi, 0 \le y \le 2\pi\).

Question 14.94

Find the absolute maximum and minimum values for the function \(f(x,y) = xy\) on the rectangle \(R\) defined by \(-1 \le x \le 1, -1 \le y\le 1\).

Question 14.95

Let \(f(x, y)=1+xy-2x+y\) and let \(D\) be the triangular region in \(\mathbb R^2\) with vertices (\(-2\), 1), (\(-2\), 5), and (2, 1). Find the absolute maximum and minimum values of \(f\) on \(D\). Give all points where these extreme values occur.

Question 14.96

Let \(f(x, y)=1+xy+x-2y\) and let \(D\) be the triangular region in \(\mathbb R^2\) with vertices (1, \(-2\)), (5, \(-2\)), and (1, 2). Find the absolute maximum and minimum values of \(f\) on \(D\). Give all points where these extreme values occur.

Question 14.97

Determine the nature of the critical points of \(f(x,y) = xy + 1/x + 8/y\).

Question 14.98

Let \(u\) be a \(C^2\) function on \(D\) which is “strictly subharmonic”; that is, the following inequality holds: \(\nabla^2u = (\partial^2u/\partial x^2) + (\partial^2u/\partial y^2) > 0\). Show that \(u\) cannot have a maximum point in \(D\backslash \partial D\) (the set of points in \(D\), but not in \(\partial D\)).

Question 14.99

Let \(u\) be a harmonic function on \(D\)—that is, \(\nabla^2 u = 0\) on \(D\backslash\partial D\)—and be continuous on \(D\). Show that if \(u\) achieves its maximum value in \(D\backslash \partial D\), it also achieves it on \(\partial D\). This is sometimes called the “weak maximum principle” for harmonic functions. [HINT: Consider \(\nabla^2(u + \varepsilon e^x), \varepsilon > 0\). You can use the following fact, which is proved in more advanced texts: Given a sequence \(\{{\bf p}_n\},\,n\,{=}\,1,2,\ldots,\) of points in a closed bounded set \(A\) in \({\mathbb R}^2\) or \({\mathbb R}^3\), there exists a point \({\bf q}\) such that every neighborhood of \({\bf q}\) contains infinitely many members of \(\{{\bf p}_n\}\).]

Question 14.100

Define the notion of a strict superharmonic function \(u\) on \(D\) by mimicking Exercise 46. Show that \(u\) cannot have a minimum in \(D\backslash \partial D\).

Question 14.101

Let \(u\) be harmonic in \(D\) as in Exercise 47. Show that if \(u\) achieves its minimum value in \(D\backslash\partial D\), it also achieves it on \(\partial D\). This is sometimes called the “weak minimum principle” for harmonic functions.

Question 14.102

Let \(\phi{:}\,\, \partial D\to {\mathbb R}\) be continuous and let \(T\) be a solution on \(D\) to \(\nabla^2T = 0,\) continuous on \(D\) and \(T = \phi\) on \(\partial D\).

• (a) Use Exercises 46 to 49 to show that such a solution, if it exists, must be unique.
• (b) Suppose that \(T(x,y)\) represents a temperature function that is independent of time, with \(\phi\) representing the temperature of a circular plate at its boundary. Can you give a physical interpretation of the principle stated in part (a)?

Question 14.103

• (a) Let \(f\) be a \(C^1\) function on the real line \({\mathbb R}\). Suppose that \(f\) has exactly one critical point \(x_0\) that is a strict local minimum for \(f\). Show that \(x_0\) is also an absolute minimum for \(f\); that is, that \(f(x) \ge f(x_0)\) for all \(x\).

• (b) The next example shows that the conclusion of part (a) does not hold for functions of more than one variable. Let \(f{:}\,\, {\mathbb R}^2 \to {\mathbb R}\) be defined by \[ f(x,y) = -y^4-e^{-x^2} + 2y^2 {\textstyle\sqrt{e^x + e^{-x^2}}}. \]

  (i)\(\quad\) Show that \((0, 0)\) is the only critical point for \(f\) and that it is a local minimum.

  (ii)\(\quad\) Argue informally that \(f\) has no absolute minimum.

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Question 14.104

Suppose that a pentagon is composed of a rectangle topped by an isosceles triangle (see Figure 14.24). If the length of the perimeter is fixed, find the maximum possible area.