Find the critical point(s) of f(x,y) = x + y^2 - e^x. Decide which are extreme points

Apr 2010
57
1
Find the critical point(s) of f(x,y) = x + y^2 - e^x. Decide which are extreme points and which are saddle points.

I got to here:
Px = 1 -e^x
Py = 2y

But now I'm stuck.

I know I'm supposed to use this rule:

d=fxxfyy-(fxy)^2

If d>0:
fxx(a,b)>0, then f(a, b) is a rel min
fxx(a,b)<0, then f(a,b) is a rel. max

If d<0, it's a saddle point.

Please help! Thank you
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Find the critical point(s) of f(x,y) = x + y^2 - e^x. Decide which are extreme points and which are saddle points.

I got to here:
Px = 1 -e^x
Py = 2y

But now I'm stuck.
At least on this one you have made an effort! The whole point of taking partial derivatives is that the critical points occur where the derivatives are 0!

Solve \(\displaystyle 1- e^x= 0\) and 2y= 0.

I know I'm supposed to use this rule:

d=fxxfyy-(fxy)^2

If d>0:
fxx(a,b)>0, then f(a, b) is a rel min
fxx(a,b)<0, then f(a,b) is a rel. max
Yes, with a and b the values you got by solving those equations.

If d<0, it's a saddle point.

Please help! Thank you
 
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mr fantastic

MHF Hall of Fame
Dec 2007
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6,768
Zeitgeist
At least on this one you have made an effort! The whole point of taking partial derivatives is that the critical points occur where the derivatives are 0!

[snip]
I hope that ! is an exclamation mark (Rofl)

For the slow minded:

0! = 1 (Rofl)
 
Apr 2010
57
1
Thanks, but I what if fxx=0?

Which in this case, it does