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Chapter 12: Q 56. (page 976)

Prove Theorem 12.42. That is, show that if f(x,y)has a local extremum at x0,y0, then x0,y0is a critical point of f.

Short Answer

Expert verified

Find the critical points and then determine local maxima and minima

Step by step solution

01

Given Information

It is given that x0,y0is local extremum of function.

02

Determining critical points

Assuming g(x)=fx,y0and h(y)=fx0,y

Fix y=y0and vary x,

g(x)=fx,y0has local maximum or minimum atx=x0

03

Simplification

for function to be single variable, f'x0=0

鈬�g(x)has local maximum or minimum at x0,g'x0=0

Also

g'x0=fxx0,y0

Same way, fix x=x0and vary y

h(y)=fx0,yhas local maxima or minima at y=y0

As per statement h'y0=0

Also, h'y0=fyx0,y0. Hence,fyx0,y0=0

Hencex0,y0is critical point off

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