# Critical Points

• Nov 1st 2008, 11:15 AM
JonathanEyoon
Critical Points
Can someone help me understand how to find critical points for this function?

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

I know i need to take the partials of fx and fy and then solve the system for critical points but i'm not certain how to do that. Thanks in advance.
• Nov 1st 2008, 12:42 PM
mr fantastic
Quote:

Originally Posted by JonathanEyoon
Can someone help me understand how to find critical points for this function?

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

I know i need to take the partials of fx and fy and then solve the system for critical points but i'm not certain how to do that. Thanks in advance.

Start by taking the partial derivatives. Helpful Hint: $f(x, y) = e^{4y - y^2} \, e^{-x^2}$. Post all your working. Then we can discuss the next step.
• Nov 1st 2008, 01:24 PM
JonathanEyoon
Quote:

Originally Posted by mr fantastic
Start by taking the partial derivatives. Helpful Hint: $f(x, y) = e^{4y - y^2} \, e^{-x^2}$. Post all your working. Then we can discuss the next step.

fx = (e^(4y-y^2))(e^(-x^2))(-2x)

fy = (e^(4y-y^2))(e^(-x^2))(4 - 2y)

I know i'm supposed to solve this system but i'm a bit confused as to how.
• Nov 1st 2008, 01:32 PM
mr fantastic
Quote:

Originally Posted by JonathanEyoon
fx = (e^(4y-y^2))(e^(-x^2))(-2x)

fy = (e^(4y-y^2))(e^(-x^2))(4 - 2y)

I know i'm supposed to solve this system but i'm a bit confused as to how.

$f_x = 0$: $0 = -2x e^{4y - y^2} e^{-x^2}$ .... (1)

$f_y = 0$: $0 = (4 - 2y) e^{4y - y^2} e^{-x^2}$ .... (2)

Is $e^{4y - y^2} = 0$ or $e^{-x^2} = 0$ possible for real value of x and y? So what do equations (1) and (2) reduce to?
• Nov 1st 2008, 01:46 PM
JonathanEyoon
nope those are not possible values since you have to take the ln of both sides and the ln0 = infinity.

Eq. 1 would reduce to 0

Eq. 2 would reduce to 2

Does this mean my candidate for a critical pt. (0,2)?
• Nov 1st 2008, 01:51 PM
mr fantastic
Quote:

Originally Posted by JonathanEyoon
nope those are not possible values since you have to take the ln of both sides and the ln0 = infinity.

Eq. 1 would reduce to 0

Eq. 2 would reduce to 2

Does this mean my candidate for a critical pt. (0,2)?

Yes. Now you have to determine its nature in the usual way.
• Nov 1st 2008, 02:10 PM
JonathanEyoon
Thanks a billion. I was just really thrown off about the value of e. Forgot something so basic (Headbang). Thanks again!
• Nov 1st 2008, 02:14 PM
mr fantastic
Quote:

Originally Posted by JonathanEyoon
Thanks a billion. I was just really thrown off about the value of e. Forgot something so basic (Headbang). Thanks again!

You're welcome. It's a pleasure to help someone who's prepared to play along and answer my questions without complaint. As a result, 95% of the solution belongs totally to you. Well done.