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Question

Find and list all critical values and identify the absolute maximum and minimum values of the function f(x,y)=x^2y over the region in the xy-plane bounded by the graphs of y=x^2 − 9 and y=x + 3 . Show your work for credit

Answer 1

Firstly plot the graphs of the given functions that define the region D and find the points of intersection You should be finding the

points A( -3, 0 ) and B( 4,7)

.

Then let's find the critical points in the interior of region D and the values of the function at these points

f_x = 2xy and f_y = x². The only critical point in D is the (0,0) and f(0,0) = 0.

Now let's turn our attention at the boundary of the region D.

The parabola y = x² -9 and the line y= x +3 , both on the interval [0,4) comprise the boundary.

1. If y = x² -9 then f(x,y) = f(x) = x²(x²-9) = x⁴−9x². then f^' (x) = 4x³−18x = 2x(2x²−9)

which has x =0, x= 3/√2, and x=- 3/√2 as critical point in the interval of interest [0,4].

The values of f at these points are 0, -81/4.

1. If y= x+3 then f(x,y) = x²(x+3)= x³ +3x² . Then f^'(x) = 3x²+6x and its critical points

x=0 and x= -2. Where the function f takes values 0 and 4.

Finally let's see the values that the function takes at the points of intersection A( -3, 0 ) and B( 4,7)

At A the function assumes the value of 0 and at B the value of 112.

If we sort the values of f we see that the function in the region D assumes its global minimum of -81/4 and its global or absolute maximum of 112 at (4,7)