Question 1

Let $f ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + d$ , for constants a, b, c, and d, with $a \neq 0$ .
The constants can be chosen in such a way that f will have a local minimum.

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Question 2

Suppose there are two electric generators that burn natural gas and whose efficiency declines with output.The energy output is an increasing (but concave down)function of fuel input.Specifically, say
Output of generator 1 is

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The answer of Suppose there are two electric generators that...

Question 3

Let $L ( x , y ) = a x + b y$ .
 There exist values of a and b so that $L ( x , y )$ takes a minimum value of 8 on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ at the point $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

Accepted Answer

The function $L(x, y) = ax + by$ is linear, and on the unit circle, it cannot have a minimum value at a specific point unless it is constant, which is not possible for a linear function on a circle.

Question 4

An exam question asks students to find the maximum of $f ( x , y )$ on the circle $g ( x , y ) = 29$ , and the gradient vectors of f and g at that point.A student gave the following

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Question 5

Find three positive numbers whose product is 11 and whose sum is a minimum.What is the minimum sum?

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The answer of Find three positive numbers whose product is...

Question 6

Let $h ( x , y , z ) = x ^ { 2 } - 2 x \sin y + z$ .Let $f ( x , y , z )$ satisfy $\nabla f ( 1,0,2 ) = \vec { i } - \vec { j } + 4 \vec { k }$ .Explain why the maximum of f subject to the constraint $h ( x , y , z ) = 3$ cannot occur at the point $( 1,0,2 )$ .

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The answer of Let \(h ( x , y ,...

Question 7

The point (0,0)is a critical point for the function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$ .What kind of critical point is it?

Accepted Answer

The function $f(x, y) = x^4 + y^4$ has a critical point at (0,0) because the first partial derivatives $f_x = 4x^3$ and $f_y = 4y^3$ both equal zero at (0,0). To determine the nature of this critical point, we examine the second partial derivatives: $f_{xx} = 12x^2$, $f_{yy} = 12y^2$, and $f_{xy} = f_{yx} = 0$. The determinant of the Hessian matrix at (0,0) is $f_{xx}f_{yy} - (f_{xy})^2 = 12*12 - 0 = 144 > 0$, and since $f_{xx} > 0$, the point (0,0) is a local minimum.

Question 8

Let $L ( x , y ) = a x + b y$ .Find values of a and b so that $L ( x , y )$ takes a maximum value of 4 on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ at the point $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

Accepted Answer

The answer of Let \(L ( x , y )...

Question 9

Let $f ( x , y ) = \sin x + \sin y + \cos ( x + y )$ in the square S bounded by $0 \leq x \leq \pi , 0 \leq y \leq \pi$ .Then $\left( \frac { \pi } { 6 } , \frac { \pi } { 6 } \right)$ is a critical point.What kind of critical point is it?

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Question 10

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Find the critical points of f.

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The answer of Let \(f ( x , y )...

Question 11

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Determine the values of k (if any)for which the critical point at (-1, 0)is a local minimum.

Accepted Answer

The answer of Let \(f ( x , y )...

Question 12

Find three numbers x, y, and z, such that $x ^ { 2 } y ^ { 3 } z = 500$ and $x + y + z$ is minimal.What is this minimal sum?

Accepted Answer

The answer of Find three numbers x, y, and z,...

Question 13

It can be shown that $( 0,0 ) , ( 0,1 / 2 ) , ( 0 , - 1 / 2 ) , ( 1,0 ) , \text { and } ( - 1,0 )$ are the critical points of the function $f ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) e ^ { - x ^ { 2 } - 4 y ^ { 2 } }$ .Which of the following are classified correctly? Select all that apply.

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Question 14

The level curves of f(x, y)are shown in the picture below.  (a)Estimate the global maximum and minimum of f on the closed triangular region D with vertices at (-1, -1), (2, -1)and (-1, 2).
(b)Find the critical point(s)of f in the interior of the region D.
(c)Find the critical point(s)of f along the boundary of D.

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The answer of The level curves of f(x, y)are shown...

Question 15

Find the maximum and minimum values of $f ( x , y , z ) = x - 4 y + 6 z$ subject to the constraint $g ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 53$ .

Accepted Answer

The answer of Find the maximum and minimum values of...

Question 16

The temperature at each point in the first quadrant is given by $T ( x , y ) = \ln ( x y ) - \frac { x } { 6 } - \frac { y } { 5 }$ .Find the hottest point in the first quadrant and determine its temperature.

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The answer of The temperature at each point in the...

Question 17

Find the maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = c$ , with $c > 0$ .Your answers may depend on c.

Accepted Answer

The answer of Find the maximum and minimum values of...

Quiz 15: Optimization- Local and Global Extrema

Let $f ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + d$ , for constants a, b, c, and d, with $a \neq 0$ . The constants can be chosen in such a way that f will have a local minimum.

Suppose there are two electric generators that burn natural gas and whose efficiency declines with output.The energy output is an increasing (but concave down)function of fuel input.Specifically, say Output of generator 1 is

Let $L ( x , y ) = a x + b y$ . There exist values of a and b so that $L ( x , y )$ takes a minimum value of 8 on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ at the point $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

An exam question asks students to find the maximum of $f ( x , y )$ on the circle $g ( x , y ) = 29$ , and the gradient vectors of f and g at that point.A student gave the following

Find three positive numbers whose product is 11 and whose sum is a minimum.What is the minimum sum?

Let $h ( x , y , z ) = x ^ { 2 } - 2 x \sin y + z$ .Let $f ( x , y , z )$ satisfy $\nabla f ( 1,0,2 ) = \vec { i } - \vec { j } + 4 \vec { k }$ .Explain why the maximum of f subject to the constraint $h ( x , y , z ) = 3$ cannot occur at the point $( 1,0,2 )$ .

The point (0,0) is a critical point for the function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$ .What kind of critical point is it?

Let $L ( x , y ) = a x + b y$ .Find values of a and b so that $L ( x , y )$ takes a maximum value of 4 on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ at the point $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

Let $f ( x , y ) = \sin x + \sin y + \cos ( x + y )$ in the square S bounded by $0 \leq x \leq \pi , 0 \leq y \leq \pi$ .Then $\left( \frac { \pi } { 6 } , \frac { \pi } { 6 } \right)$ is a critical point.What kind of critical point is it?

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Find the critical points of f.

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Determine the values of k (if any)for which the critical point at (-1, 0)is a local minimum.

Find three numbers x, y, and z, such that $x ^ { 2 } y ^ { 3 } z = 500$ and $x + y + z$ is minimal.What is this minimal sum?

It can be shown that $( 0,0 ) , ( 0,1 / 2 ) , ( 0 , - 1 / 2 ) , ( 1,0 ) , \text { and } ( - 1,0 )$ are the critical points of the function $f ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) e ^ { - x ^ { 2 } - 4 y ^ { 2 } }$ .Which of the following are classified correctly? Select all that apply.

Find the maximum and minimum values of $f ( x , y , z ) = x - 4 y + 6 z$ subject to the constraint $g ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 53$ .

The temperature at each point in the first quadrant is given by $T ( x , y ) = \ln ( x y ) - \frac { x } { 6 } - \frac { y } { 5 }$ .Find the hottest point in the first quadrant and determine its temperature.

Find the maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = c$ , with $c > 0$ .Your answers may depend on c.

Quiz 15: Optimization- Local and Global Extrema

Let f(x,y)=ax2+bx−y2+cy+df ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + df(x,y)=ax2+bx−y2+cy+d , for constants a, b, c, and d, with a≠0a \neq 0a=0 . The constants can be chosen in such a way that f will have a local minimum.

Suppose there are two electric generators that burn natural gas and whose efficiency declines with output.The energy output is an increasing (but concave down)function of fuel input.Specifically, say Output of generator 1 is

An exam question asks students to find the maximum of f(x,y)f ( x , y )f(x,y) on the circle g(x,y)=29g ( x , y ) = 29g(x,y)=29 , and the gradient vectors of f and g at that point.A student gave the following

Find three positive numbers whose product is 11 and whose sum is a minimum.What is the minimum sum?

The point (0,0) is a critical point for the function f(x,y) =x4+y4f ( x , y ) = x ^ { 4 } + y ^ { 4 }f(x,y) =x4+y4 .What kind of critical point is it?

Let f(x,y)=kx3−3kx+y2f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }f(x,y)=kx3−3kx+y2 where k ≠\neq= 0.Find the critical points of f.

Let f(x,y)=kx3−3kx+y2f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }f(x,y)=kx3−3kx+y2 where k ≠\neq= 0.Determine the values of k (if any)for which the critical point at (-1, 0)is a local minimum.

Find three numbers x, y, and z, such that x2y3z=500x ^ { 2 } y ^ { 3 } z = 500x2y3z=500 and x+y+zx + y + zx+y+z is minimal.What is this minimal sum?

Find the maximum and minimum values of f(x,y,z)=x−4y+6zf ( x , y , z ) = x - 4 y + 6 zf(x,y,z)=x−4y+6z subject to the constraint g(x,y,z)=x2+y2+z2=53g ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 53g(x,y,z)=x2+y2+z2=53 .

The temperature at each point in the first quadrant is given by T(x,y)=ln⁡(xy)−x6−y5T ( x , y ) = \ln ( x y ) - \frac { x } { 6 } - \frac { y } { 5 }T(x,y)=ln(xy)−6x​−5y​ .Find the hottest point in the first quadrant and determine its temperature.

Find the maximum and minimum values of f(x,y)=x2+7xy+y2f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }f(x,y)=x2+7xy+y2 subject to the constraint x2+y2=cx ^ { 2 } + y ^ { 2 } = cx2+y2=c , with c>0c > 0c>0 .Your answers may depend on c.

Let $f ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + d$ , for constants a, b, c, and d, with $a \neq 0$ . The constants can be chosen in such a way that f will have a local minimum.

An exam question asks students to find the maximum of $f ( x , y )$ on the circle $g ( x , y ) = 29$ , and the gradient vectors of f and g at that point.A student gave the following

The point (0,0) is a critical point for the function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$ .What kind of critical point is it?

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Find the critical points of f.

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Determine the values of k (if any)for which the critical point at (-1, 0)is a local minimum.

Find three numbers x, y, and z, such that $x ^ { 2 } y ^ { 3 } z = 500$ and $x + y + z$ is minimal.What is this minimal sum?

Find the maximum and minimum values of $f ( x , y , z ) = x - 4 y + 6 z$ subject to the constraint $g ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 53$ .

The temperature at each point in the first quadrant is given by $T ( x , y ) = \ln ( x y ) - \frac { x } { 6 } - \frac { y } { 5 }$ .Find the hottest point in the first quadrant and determine its temperature.

Find the maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 7 x y + y ^ { 2 }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = c$ , with $c > 0$ .Your answers may depend on c.