Find the Critical Point Of $f ( x , y ) = x ^ { 2 } y e ^ { - \left( x ^ { 2 } + 25 y ^ { 2 } \right) }$

question 1

Essay

Find the critical point of $f ( x , y ) = x ^ { 2 } y e ^ { - \left( x ^ { 2 } + 25 y ^ { 2 } \right) }$ Do this by setting $t = - \left( x ^ { 2 } + 25 y ^ { 2 } \right)$ and optimizing $f ( x , y , t ) = x ^ { 2 } y e ^ { t }$ subject to the constraint $t + x ^ { 2 } + 25 y ^ { 2 } = 0$ What are the global maximum and minimum values of f? Give your answer to 4 decimal places.

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Definitions:

Mean

The arithmetic average of a set of numbers, representing the central location within that dataset.

Standard Normal

Refers to a normal distribution with a mean of zero and a standard deviation of one, serving as a basis for comparing other normal distributions.

Standard Normal

A distribution that is standard and normal with its average at zero and its variability defined by a standard deviation of one.

Area

The measure of the extent of a two-dimensional surface or shape, or the extent of a land surface within specific boundaries.