Find and classify the critical points of the given equation.

Question

Find and Classify the critical point of

f(x,y) =192x³+y²-4xy² on the triangle with vertices (0,0), (4, 2) and (-2,2).

Answer 1

f(x,y) =192x³+y²-4xy²

f_x=384x-4y²,f_xx=384

f_y=2y-8xy, f_yy=2-8x

f_xy=f_yx=-8y

f_x=0 and f_y=0 when (x,y) =(0,0) (and when x=1/4 and y=±√24)

For the region described, only (x,y)= (0,0) applies so it is a critical point

using f_xx,f_xy,f_yx,f_yy in Hessian Determinant at point (0,0), find its value>0 and f_xx>0

meaning the point (0,0) is a minimum