Exam 15: Optimization- Local and Global Extrema

The contour diagram of f is shown below.Which of the points A, B, C, D, and E appear to be critical points? Select all that apply.

The Perfect House company produces two types of bathtub, the Hydro Deluxe model and the Singing Bird model.The company noticed that demand and prices are related.In particular, for Hydro Deluxe: demand = 1900 - price of Hydro Deluxe + price of Singing Bird for Singing Bird: demand = 1450 + price of Hydro Deluxe -2(price of Singing Bird). The costs of manufacturing the Hydro Deluxe and Singing Bird are $500 and $300 per unit respectively.Determine the price of each model that gives the maximum profit.

Level curves of f(x, y)are shown in the figure below.(Darker shades indicate regions with lower levels.) Is the point (-1, 2)a local maximum, a local minimum, or a saddle point of f, or is it none of these?

Suppose that $f ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } - x$ Find the minimum value of the function f when (x,y)is constrained to lie on or inside the triangle with vertices (0,-2), (0,1), and (1,-2).Give your answer to 4 decimal places.

Suppose that (1, -4)is a critical point of a smooth function f(x, y)with $f ( 1 , - 4 ) = - 1 , f _ { x x } ( 1 , - 4 ) = 2 , f _ { x y } ( 1 , - 4 ) = 2 , f _ { y y } ( 1 , - 4 ) = - 3$ Find the quadratic approximation of f at (1, -4).

Find the critical points of $f ( x , y ) = x ^ { 2 } y e ^ { - \left( x ^ { 2 } + 9 y ^ { 2 } \right) }$ Do this by setting $t = - \left( x ^ { 2 } + 9 y ^ { 2 } \right)$ and optimizing $f ( x , y , t ) = x ^ { 2 } y e ^ { t }$ subject to the constraint $t + x ^ { 2 } + 9 y ^ { 2 } = 0$

The following results are obtained when optimizing f(x, y)subject to the constraint g(x, y)= 39.The maximum value is f(5, 7)= 42, the Lagrange multiplier $\lambda$ = 3 (when $\nabla f = \lambda \nabla g$ )and $\nabla f = 3 \vec { i } - 3 \vec { j }$ .If the constraint condition is changed to g(x, y)= 40, what will be new maximum value of f(x, y)?

Consider the four points A = (1, 0), B = (2, 2), C = (3, 5)and D = (4, 3)in the xy-plane. Find a and b in the line of best fit y = ax + b for these points. (The line of best fit minimizes the sum of the squares of the vertical distances from each point to the line.)

The point (-2, 1)is a critical point of $g ( x , y ) = 4 x ^ { 3 } - 96 x y + 48 x y ^ { 2 }$ Classify it either as a local minimum, local maximum, or saddle point.

Determine three positive numbers x, y, z that maximize $x ^ { 3 } y ^ { 4 } z ^ { 5 }$ under the condition x + y + z = 17.

If C is a circle in the plane, and if f(x, y)is differentiable and is not constant when constrained to C, then there must be at least one point on C where $\operatorname { grad } f$ is perpendicular to C.

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 $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

The following results are obtained when optimizing f(x, y)subject to the constraint g(x, y)= 35.The maximum value is f(5, 7)= 39, the Lagrange multiplier $\lambda$ = 4 (when $\nabla f = \lambda \nabla g$ )and $\nabla f = 2 \vec { i } - 5 \vec { j }$ .What is g(5, 7)?

A company manufactures a product using x, y and z units of three different raw materials.The quantity produced is given by the function $Q = 60 x ^ { \frac { 1 } { 3 } } y ^ { \frac { 1 } { 4 } } z ^ { \frac { 2 } { 5 } }$ .Suppose the cost of the materials per unit is $20, $25 and $75 respectively. Find the maximum production if the budget is limited to $6000.

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 $f ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + d$ , 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

Suppose that $f ( x , y ) = 4 x ^ { 2 } + 4.0000 y ^ { 2 } - x$ Find and classify (as local maxima, minima, or saddle points)all critical points of f.

Consider the function $f ( x , y ) = x ^ { 4 } + 2 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 7 x y ^ { 3 } - y ^ { 4 } + 3$ Check that (0,0)is a critical point of f and classify it as a local minimum, local maximum or saddle point.

Let $\vec { v } = 3 \vec { i } + a \vec { j } + b \vec { k }$ be a vector in space with a, b > 0. Compute the cross product $\vec { v } \times ( 3 \vec { j } + \vec { k } )$ and then use the result and the Lagrange Multiplier method to find the values of a and b such that the magnitude of the cross product $\| \vec { v } \times ( 3 \vec { j } + \vec { k } ) \|$ is the largest with $\| \vec { v } \| = 19 .$