Q. 6 - Calculus

Question

Suppose f is a function that is defined and continuous on a closed interval I. Will the endpoints of I always be local extrema of f ? Will f necessarily have a global maximum or minimum in the interval I? Justify your answers.

Step-by-Step Solution

Answer

The reason has been explained.

Step 1. Given information.

We have to explain that : Suppose f is a function that is defined and continuous on a closed interval I. Will the endpoints of I always be local extrema of f ? Will f necessarily have a global maximum or minimum in the interval I? 

Step 2. Explanation

According to the extreme value theorem,

If f is a continuous function on a closed interval I, then $f$ has a global maximum and a global minimum inside I.

It means that there exists a number K such that f(K)≤ f(x) for all x inside I. ( f(K) is the global maximum), and also there exists a number K such that f(K) ≤ f(x) for all x inside I ( f(K) is the global minimum). 

It is not necessary for a function to have a global maximum or minimum, but the extreme Value Theorem tells us that a continuous function inside a closed interval must have a global maximum and a global minimum.