How to determine image of function

Question

I have a continuous function $$f:[a,b] \to \mathbb R,$$ for $a,b \in \mathbb R,a < b$. $f$ is differentiable in $(a,b)$.

How can I determine The image of $f(|a,b|)$?

Answer 1

If $f: [a,b] \to \mathbb{R}$ is a continuous function, then the Intermediate Value Theorem and Extreme Value Theorem say that $f$ attains an absolute maximum and absolute minimum value on $[a,b]$. In fact the image of $f$ will be the closed interval from this min. value to the max. value.

If $f$ is also differentiable on $(a,b)$, then we have the so-called Closed Interval Method that says

1. Find all critical numbers of $f$ (that is all numbers $c$ where $f'(c) = 0$ or where $f'(c)$ doesn't exist)
2. Let $m$ be the smallest number of $f(a), f(b)$, and $f(c)$ where $c$ is a critical number. Let $M$ be the largest from this list of numbers.

Then the image of $f$ is $[m,M]$.