Exam 15: Optimization- Local and Global Extrema

What do the second derivatives tell you about the graph of $f ( x , y ) = 7 x ^ { 2 } + 1000 x y + 7 y ^ { 2 } - 3 x - 12 y + 18 ?$ Select all that apply.

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

Suppose the quantity, q, of a good produced depends on the number of workers, w, and the amount of capital, k, invested and is represented by the Cobb-Douglas function $q = 6 w ^ { \frac { 3 } { 4 } } k ^ { \frac { 1 } { 4 } }$ .In addition, labor costs are $20 per worker and capital costs are $20 per unit, and the budget is $3680.Using Lagrange multipliers, find the optimum number of units of capital.

Consider the function $f ( x , y ) = x + y + \frac { 4 } { x } + \frac { 9 } { y }$ Determine all the local maximum, minimum and saddle points in the region $x > 0 , y < 0$

Suppose that $f ( x , y ) = x ^ { 2 } + 3 x y + y ^ { 2 }$ Find a normal vector to the tangent plane of f at the point (1, 1).Select all that apply.

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

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

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

Level curves of f(x, y)are shown in the figure below.(Darker shades indicate regions with lower levels.) Determine if $f _ { y } ( 1,0 )$ is positive, negative or zero.

The owner of a jewelry store has to decide how to allocate a budget of $540,000.He notices that the earnings of the company depend on investment in inventory x₁ (in thousands of dollars)and expenditure x₂ on advertising (in thousands of dollars)according to the function $f ( x , y ) = 6 x _ { 1 } ^ { 2 / 3 } x _ { 2 } ^ { 1 / 3 }$ How should the owner allocate the $540,000 between inventory and advertising to maximize his profit?

A zoo is designing a giant bird cage consisting of a cylinder of radius r feet and height h feet with a hemisphere on top (no bottom).The material for the hemisphere costs $20 per square foot and the material for the cylindrical sides costs $10 per square foot; the zoo has a budget of $5120.Find the values of r and h giving the birds the greatest space inside assuming the zoo stays within its budget.

Let $h ( x , y ) = x ^ { 3 } + y ^ { 3 } + 9 x y + 6$ Determine all local maxima, minima, and saddle points.Are the local extrema also global extrema?

Find a point on the surface x-yz = 14 that is closest to the origin.

Suppose that (0, -2)is a critical point of a smooth function f(x, y)with $f ( 0 , - 2 ) = - 2 , f _ { x x } ( 0 , - 2 ) = - 1 , f _ { x y } ( 0 , - 2 ) = 2 , f _ { y y } ( 0 , - 2 ) = - 5$ What can you conclude about the behavior of the function f near (0, -2)?

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 $( 1,0,2 )$ .

Suppose that $f ( x , y ) = x ^ { 2 } + 3 x y + y ^ { 2 }$ Find an equation of the tangent plane to the graph of f at the point (2, 2).

A company manufactures a product using x, y and z units of three different raw materials.The quantity produced is given by The production is described 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, $15 and $24 respectively. (a)Find the cheapest way to produce 6300 units of the product. (b)Find the value of $\lambda$ in $\nabla f = \lambda \nabla Q$ and interpret this value.