Question

Please check my answers and let me know if I did something wrong. Thank you!

Find the partial derivative of x, and the partial derivative of y, then the partial derivative of x (1,-1), and the partal derivative of y (1,-1).

f(x,y) = x^4 y^2 -x

here is what I got
@f/@x = 3x^3 y^2 -1
@f / @y = x^4 2y
@f / @x = 3(1)^3 (-1)^2 -1 = 2
@f / @y = (1)^4 2(-1) = -2


f(x,y) = e^(2x+y)
@f/@x = 2e^(2x+y)
@f / @y = e^(y+2x)
@f / @x = 2e
@f / @y = e

Locate and classify the critical points the functions.
g(x,y) = -x^2 -2xy + y^2 + x -4y
g(x) = -2x -2y +1
g(y) = -2x + 2y -4
set them equal to zero?
g(x) = -2x -2y +1 = 0
g(y) = -2x + 2y -4 = 0
Here is what I got.
(-3/4, 5/4)

f(x,y) xe^y
f(x) = e^y
f(y) = e^(y) x
x=0

Answer 1

Looks good, except for the last one. It has no critical points, since Fx is never zero.

Note that the critical points are where both Fx and Fy are zero.