   Chapter 14, Problem 7RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has a local minimum at (a, b) and f is differentiable at (a, b) then ∇f(a, b) = 0.

To determine

Whether the statement “If f has a local minimum at (a,b) and f is differentiable at (a,b) , then f(a,b)=0 ” is true or false.

Explanation

Theorem used:

If f has a local minimum at (a,b) and the first-order partial derivatives of f exist there, , then fx(a,b)=0 and fy(a,b)=0 .

Reason:

From the theorem, f(x,y) has local minimum and differentiable.

Then, fx(a,b)=0 and fy(a,b)=0 .

The value of f(a,b) is,

f(a,b)=

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