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Math Help - Critical Points

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by JonathanEyoon View Post
    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.
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    Quote Originally Posted by mr fantastic View Post
    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.
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    Quote Originally Posted by JonathanEyoon View Post
    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?
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  5. #5
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    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)?
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  6. #6
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    Quote Originally Posted by JonathanEyoon View Post
    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.
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  7. #7
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    Thanks a billion. I was just really thrown off about the value of e. Forgot something so basic . Thanks again!
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  8. #8
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    Quote Originally Posted by JonathanEyoon View Post
    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.
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