# Thread: Critical Points

1. ## 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.

2. 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: $\displaystyle f(x, y) = e^{4y - y^2} \, e^{-x^2}$. Post all your working. Then we can discuss the next step.

3. Originally Posted by mr fantastic
Start by taking the partial derivatives. Helpful Hint: $\displaystyle f(x, y) = e^{4y - y^2} \, e^{-x^2}$. 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.

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

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

Is $\displaystyle e^{4y - y^2} = 0$ or $\displaystyle e^{-x^2} = 0$ possible for real value of x and y? So what do equations (1) and (2) reduce to?

5. 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)?

6. 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.

7. Thanks a billion. I was just really thrown off about the value of e. Forgot something so basic . Thanks again!

8. Originally Posted by JonathanEyoon
Thanks a billion. I was just really thrown off about the value of e. Forgot something so basic . 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.