Find the local maximum and minimum values and saddle point(s) of the function. I

Question

Find the local maximum and minimum values and saddle point(s) of the function. If you have three dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter NONE in any unused answer blanks.)
f(x, y) = 9e^(4y) - x^2 - y^2
maximum
f(______, _____) =
(smaller x value)


f( ______, ______) =
(larger x value)


minimum
f( ______, _____) =
(smaller x value)


f( _______, _____) =
(larger x value)


saddle points
(_____, _____) (smallest x value)
( _____, _____) (largest x value)

Explanation / Answer

df/dx = 9 * (-2x) * e^(4y - x^2 - y^2) df/dy = 9 * (4 - 2y) * e^(4y - x^2 - y^2). Setting df/dx and df/dy = 0, we get the critical point (0,2). (Note that e^(4y - x^2 - y^2) is never 0.) As for classifying this point: f_xx = 9 * (-2) * e^(4y - x^2 - y^2) + 9 * (-2x)^2 * e^(4y - x^2 - y^2) ==> f_xx (0,2) = -18 e^4 f_yy = 9 * (-2) * e^(4y - x^2 - y^2) + 9 * (4 - 2y)^2 * e^(4y - x^2 - y^2) ==> f_yy (0,2) = -18e^4 f_xy = 9 * (-2x) * (4 - 2y) * e^(4y - x^2 - y^2) ==>f_(xy) (0,2) = 0. So, D(0,2) = [(f_xx)(f_yy) - (f_xy)^2](0,2) = 324e^8 > 0 while f_xx (0,2) < 0. Thus, we have a local maximum at (x,y) = (0,2).

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