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Chapter 12: Q. 23 (page 985)

Let f(x,y)be a differentiable function such that 鈭�f(x,y)鈮�0for every point in the domain of f, and let Rbe a closed, bounded subset of role="math" localid="1649887954022" 鈩�2.Explain why the maximum and minimum of f restricted to Roccur on the boundary ofrole="math" localid="1649888770915" R.

Short Answer

Expert verified

As the function does not have any critical point so the maximum and minimum of f are restricted to Roccur on the boundary ofR.

Step by step solution

01

Step 1. Given information.   

The function f(x,y)is differentiable and 鈭�f(x,y)鈮�0.

Ris a closed and bounded subset of 鈩�2.

02

Step 2. Explanation

At critical points of a function, 鈭�f(x,y)=0but here 鈭�f(x,y)鈮�0.

so function does not have any critical points.

extreme value theorem state that the function of two variables defined on a closed and bounded set has maximum and minimum on the closed and bounded set.

As the function does not have any critical point so the maximum and minimum of f are restricted to Roccur on the boundary of role="math" localid="1649888808133" R.

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