Proof of stationary points of 3D function

[Lucy Yeats]

Homework Statement

Show that f(x,y)=g(x,y)h(x,y) is stationary if and only if:
g=0 and h=0
OR
df/dx=0 and (dg/dx)(dh/dy)=(dg/dy)(dh/dx)
(All the d's in the line above should be curly d's for partial derivatives.)

Homework Equations

The Attempt at a Solution

I tried expressing df as a total derivative then setting this equal to zero, but I wasn't getting anywhere.

 

[Lucy Yeats]

Any suggestions would be great. :)

 

[Dick]

f is stationary if df/dx=0 and df/dy=0 (the d's being partial derivatives). g=0 and h=0 are certainly sufficient to show that. Just use the product rule. Where did you go from there? Can you show your work? Use that one of g or h must be nonzero.