Concept preview

What you will be able to do

1. 01State and apply the Candidates Test: to locate the absolute (global) extrema of a continuous function on a closed interval $[a,\,b]$, evaluate $f$ at every critical number inside $(a,\,b)$ and at both endpoints, then compare those output values.
2. 02Organise the work as a table of candidate values, and read off the absolute maximum (largest value) and absolute minimum (smallest value), naming both the extreme value and the $x$ where it occurs.
3. 03Avoid the two classic slips — forgetting to test the endpoints, and reporting the $x$-location instead of the actual extreme value $f(x)$.

Suppose $f$ is continuous on a closed interval $[a,\,b]$. Where could its highest and lowest points possibly sit? Only three kinds of place. They could sit at an interior point where the tangent is horizontal or the derivative fails to exist — that is, at a critical number, where $f'(x)=0$ or $f'(x)$ is undefined. Or they could sit at one of the two endpoints $x=a$ or $x=b$, where the interval simply stops and the curve is not free to keep climbing or falling. There is nowhere else for a global extremum to hide, because at any interior point with $f'(x)\neq 0$ the function is strictly rising or falling, so you could always step a little to one side and find a higher or lower value.