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Consider the following function.

$g(x, y)=e^{−8x^2−6y^2+24y}$

(a) Find the critical point of g.

(b) Using your critical point in (a), find the value of D(a, b) from the Second Partials test that is used to classify the critical point.

(c) Use the Second Partials test to classify the critical point from (a).

For C, the options are either

Saddle Point, Relative Minimum, Relative Maximum, or Inconclusive

I could really used some help finding the critical point. I separated them into $f_x$ and $f_y$ and set them equal to 0 and got (24,0). I'm not sure if this answer is right and what I am supposed to do to find b.

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    $\begingroup$ If $f(x) = e^x$ is a strictly increasing function with nonzero derivative everywhere, the set of critical points for $f{g(x)]$ is the same as the set of critical points for $g(x)$. So you might just answer your question with respect to the function $-8x^2-6y^2+24y$. $\endgroup$ – mlc Mar 13 '17 at 7:43
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Without derivatives:

$g(x,y)=\exp(-8x^2-6(y-2)^2+24 ) \le \exp(24)=g(0,2)$

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  • $\begingroup$ Thank you! but do you know how I should plug it into the Hessian matrix to solve for b? $\endgroup$ – Maggie.D Mar 13 '17 at 19:20
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Your answer is not right ! We have

$g_x(x,y)=g(x,y)(-16x)=0 \iff x=0$

and

$g_y(x,y)=g(x,y)(-12y+24)=0 \iff y=2$.

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  • $\begingroup$ Why the downvote ??????? $\endgroup$ – Fred Mar 14 '17 at 5:13

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