The Function $f ( x , y ) = e ^ { - ( x - a ) ^ { 2 } - ( y - b ) ^ { 2 } }$

question 20

Essay

The function $f ( x , y ) = e ^ { - ( x - a ) ^ { 2 } - ( y - b ) ^ { 2 } }$ where a and b are constants is sometimes referred to as a "bump function" and is used to construct functions which take on maximum values at certain points.Show that f(x, y)has a maximum at (a, b).

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Definitions:

Phenylketonuria

A genetic disorder leading to the buildup of phenylalanine in the body, which can cause intellectual disability and other health issues without treatment.

Tay-Sachs Disease

A rare, inherited disorder that destroys nerve cells in the brain and spinal cord, marked by absence of a specific enzyme leading to progressive neurological deterioration.

Sequential Processing

A method of processing information in which the brain analyzes data in a linear, step-by-step manner.

Simultaneous Processing

The cognitive ability to process several components of information at once, integrating them into a comprehensive understanding.