The figure shows a contour graph for a function f in blue with a constraint func

Question

The figure shows a contour graph for a function f in blue with a constraint function g = 2 in black. Ignoring the constraint g = 2, locate the optimal point of f and, if one exists, classify it as a relative maximum point or a relative minimum point. (If an answer does not exist, enter DNE.) (k, h, f(k, h)) = (_____) Estimate any optimal points for the system {optimize f subject to g = 2. Classify each constrained optimal point as a maximum or a minimum. (Order your answers from smallest to largest k.) (k, h, f(k, h)) = (_____) (k, h, f(k, h)) = (_____)

Explanation / Answer

(a) From the given contour graph of function f, we can see that we will have higher values of f(x,y) for larger values of h and k. Therefore, the value of f(k,h) will keep rising and is unbounded. Therfore, relative maximum point of the function f(x,y) does not exist. However, if we do not consider g(x,y) as a constraint, then we can see that as we reduce the values of h and k, the f(k,h) approaches to zero. Therefore, relative minima of f(x,y) is 0.

Hence (k,h,f(k,h)) = (0,0,0)

(b) Now, if we take into consideration the constraint g=2, then we can see that we have two points where f(x,y) meets g=2.

These two points will be the points of optimization. Points are, (-1,1) and (1,1). At both these points, the maximum value of the function is 2. Therefore,

(k,h,f(k,h)) = (-1,1,2) (Maximum)

(k,h,f(k,h)) = (1,1,2) (Maximum)

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