Chapter 4, Problem 1RWDT

To determine

To explain: In a closed interval, a continuous function having only two critical points must have absolute maxima and absolute minima.

Answer to Problem 1RWDT

The extreme value theorem is explained.

Explanation of Solution

Given information: A continuous function has only two critical points over a closed interval.

Calculation:

Extreme value theorem: According to this theorem, if a function is continuous over a closed interval $\left[a,b\right]$ then, the function has both maximum and minimum value on the interval.

• Absolute extremes are the points where the derivative of continuous function is either zero or does not exist.

  $f\text{'}(c)=0$

• If there are only two critical points then, it is necessary that at one absolute maximum will occur and at the another critical point there will be absolute minimum.