Question

Localate all critical points for the function z = x² - 4xy/(y²+1). Hence classify them as local maximum, local minimum or saddle points if exist

Answer 1

I don't have time to do this question, but here's how you do it
say that function f is x² - 4xy/(y²+1)
find derivative of that function with respect to x, call that fx
then, with respect to y, call that fy

set both equal to zero and solve for local mins and maxs.
these are the critical points of the function

then using the second derivative test, you can find local min and max
find the derivative of fx with respect to x and y, calling them, fxx and fxy
find the derivative of fy with respect to y, calling it, fyy
apply second derivative test
D = fxx(a,b) * fyy(a,b) - [fxy(a,b)]^2
(where (a,b) are the critical points that you've found)

if D > 0 and fxx(a,b) is > 0, then f(a,b) is local min
if D > 0 and fxx(a,b) is < 0, then f(a,b) is local max
if D = 0 saddle point.

good luck