Exam 15: Optimization- Local and Global Extrema

Find the critical point of $f ( x , y ) = x ^ { 2 } y e ^ { - \left( x ^ { 2 } + 25 y ^ { 2 } \right) }$ Do this by setting $t = - \left( x ^ { 2 } + 25 y ^ { 2 } \right)$ and optimizing $f ( x , y , t ) = x ^ { 2 } y e ^ { t }$ subject to the constraint $t + x ^ { 2 } + 25 y ^ { 2 } = 0$ What are the global maximum and minimum values of f? Give your answer to 4 decimal places.

Given that the quadratic Taylor polynomial of f at (4, 5)is $p ( x , y ) = - 2 + x ^ { 2 } + y ^ { 2 } - 4 x + 12 y$ decide whether $( 4,5 )$ is a critical point.If so, identify what sort of critical point it is.

Consider the function $f ( x , y ) = e ^ { 1 - x ^ { 2 } - 2 y ^ { 2 } }$ Determine all local maxima, minima and saddle points of f. Does f have a global maximum?

Suppose that $f ( x , y ) = x ^ { 2 } + 4 x y + y ^ { 2 }$ Find and classify the critical point(s)as local maxima, local minima, or saddle points.

Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 4 y ^ { 2 } - b x$ , where a, b are any positive numbers with a $\gt$ 4. Find the minimum value of a such that the critical point will be a saddle point.

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.

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.

A company has $250,000 to spend on labor and raw materials.Let L be the quantity of labor and R be the quantity of raw materials.The production output P of the company is cRL (here c is a positive constant).Suppose that each unit of labor costs $6000 and the unit price of raw materials is $2000. Find the ratio of R to L that maximizes P.

Suppose that you want to find the maximum and minimum values of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint x + 4y = 3. Use the method of Lagrange multipliers to find the exact location(s)of any extrema.

Find all the critical points of $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 4 y$ and classify each as maximum, minimum, or saddle point. Select all possible choices.

Let $f ( x , y ) = x - y$ .The constraint g(x, y)= 3 is sketched in the picture below. In the picture, locate the point where f will have a global maximum subject to the constraint g(x, y)= 3.

Use Lagrange multipliers to find the minimum value of 4xy on the circle $x ^ { 2 } + y ^ { 2 } = 4$ .

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?

(a)Let $f ( x , y ) = x ^ { 2 } - 4 x y + 2 y ^ { 2 }$ .Find the maximum and minimum values of f on the curve $x ^ { 2 } + 2 y ^ { 2 } = 1$ (b)Use the results of part (a)to find the maximum and minimum values of $h ( x , y ) = e ^ { x ^ { 2 } - 4 x y + 2 y ^ { 2 } }$ on the curve $x ^ { 2 } + 2 y ^ { 2 } = 1$ Explain your work.

The contour diagram of f is shown below. Find and classify the critical points. Describe possible gradient vectors of f at points C, D and E.

Describe the shape of the graph of $f ( x , y ) = 3 x ^ { 2 } + 3 y ^ { 2 } - 3 x - 12 y + 18$

Consider the diagram shown below, which shows gradient vectors of a function f(x, y). Which is less: f(A)or f(C)?

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?

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

The function $f ( x , y ) = e ^ { - ( x - a ) ^ { 2 } - ( y - b ) ^ { 2 } }$ where a and b are constants is sometimes referred to as a "bump function" and is used to construct functions which take on maximum values at certain points.Show that f(x, y)has a maximum at (a, b).