Exam 15: Optimization- Local and Global Extrema

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 saddle point of $f ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) e ^ { x / 2 }$ .

The Lagrange multipliers needed to find the maximum and minimum values of $f ( x , y ) =$ 8xy on the circle $x ^ { 2 } + y ^ { 2 } = 25$ is $\lambda = \pm 4$ , where $\nabla f = \lambda \nabla g$ .Estimate the maximum and minimum values of 8xy subject to the constraint $x ^ { 2 } + y ^ { 2 } = 30.25$ .

Find the maximum and minimum values of the function $f ( x , y ) = x ^ { 2 } + 6 x y + y ^ { 2 }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } \leq 32$ .

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.

Is (0, 0)a critical point of the following function? $f ( x , y ) = \cos x \cos y$

A company has two manufacturing plants which manufacture the same item.Suppose the cost function is given by $C \left( q _ { 1 } , q _ { 2 } \right) = 4 q _ { 1 } ^ { 2 } + q _ { 1 } q _ { 2 } + q _ { 2 } ^ { 2 }$ where q₁ and q₂ are the quantities (measured in thousands)produced in each plant.The total demand q₁ + q₂ is related to the price, p, by $p = 110 - 0.5 \left( q _ { 1 } + q _ { 2 } \right)$ How much should each plant produce in order to maximize the company's profit?

Find the critical points of $f ( x , y ) = x ^ { 3 } - 9 x y + y ^ { 2 }$ and classify each as maximum, minimum or saddle.

Without calculating the discriminant, explain using the contour diagram for $f ( x , y ) = x ^ { 3 } + 3 x ^ { 2 } y$ why f has a saddle point at (0, 0).

Let $h ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 x y + 5$ Which figure best represents the level curves of this function?

A coffee company sells three brands of coffee.Brand A costs p₁ dollars per can, brand B costs p₂ dollars per can, and brand C costs p₃ dollars per can.The demand (in hundreds of cans)depends on the prices as follows: demand for brand A $= 200 - p _ { 1 }$ _, demand for brand B $= 300 - 2 p _ { 2 }$ _, demand for brand C $= 400 - 3 p _ { 3 }$ . The company can produce 69,000 cans.What selling prices optimize the total revenue?

The function f(x, y)has a local maximum at (-1, 1). What can you say (if anything)about the values of $f _ { x y } ( - 1,1 )$ ?

Determine the nature of the critical points of the function $f ( x , y ) = 3 x ^ { 5 } y + x y ^ { 5 } + x y$

The function $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 6 y$ has a saddle point at (-1, 12).Which of the following is a sketch of the level curves of f near this point?

Find a and b so that $f ( x , y ) = a x ^ { 2 } + b x y + y ^ { 2 }$ has a critical point at (1, 6).

Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 2 y ^ { 2 } - b x$ , where a, b are any numbers with a $\neq$ 2. Find the critical point of f.(Express your answer in terms of the constants a and b.)

The Green Leaf Bakery makes two types of chocolate cakes, Delicious and Extra Delicious.Each Delicious requires 0.1 lb of European chocolate, while each Extra Delicious requires 0.2 lb.Currently there are only 233 lb of chocolate available each month.Suppose the profit function is given by: $p ( x , y ) = 151 x - 0.2 x ^ { 2 } + 200 y - 0.1 y ^ { 2 }$ where x is the number of Delicious cakes and y is the number of Extra Delicious cakes that the bakery produces each month. (a)How many of each cake should the bakery produce each month to maximize profit? (b)What is the value of $\lambda$ in part (a)(if $\nabla p = \lambda \nabla g$ )? What does it mean? (c)It will cost $19.00 to get an extra pound of European chocolate.Should the bakery buy it?

Find all the critical points of the function $f ( x , y ) = x y + \frac { 8 } { x ^ { 2 } } + \frac { 8 } { y ^ { 2 } }$ Classify these critical points as local maxima, local minima, or saddle points.

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.