Find the absolute extrema of the function 2x 2xy y2 on the

Find the absolute extrema of the function 2x - 2xy + y^2 on the region in the xy plane bounded by the graphs of y = x^2 and y = 4. No work, no credit. Messy work, no credit. Box your answer.

Solution

The maxima and minima points are given by f_x =0 and f_y =0, where f(x,y)=2x-2xy+y² .

f_x = 2- 2y = 0. It implies y = 1.

f_y = -2x+ 2y = 0. It implies x = y. For y = 1, x = 1. So that stationary point is (1,1) of f(x,y) and it lies inside the curve y=x² and y = 4.

A= f_xx = 0, B=f_xy = - 2 and C = f_yy = 2.

A C - B² = -4 > 0 . Thus the point (1,1) is neither maximum nor minimum.