Concept preview

What you will be able to do

1. 01State the Extreme Value Theorem — a function continuous on a CLOSED interval $[a,\,b]$ attains both an absolute maximum and an absolute minimum on that interval — and explain why closedness and continuity are both required.
2. 02Find the critical numbers of a function (the interior inputs where $f'(x)=0$ or $f'(x)$ is undefined) and distinguish local (relative) extrema from global (absolute) extrema.
3. 03Explain why a critical point is only a CANDIDATE for an extremum, using $f(x)=x^3$ at $x=0$, and locate absolute extrema on a closed interval with the candidates test.

The Extreme Value Theorem (EVT) makes a clean promise: if $f$ is continuous on a closed interval $[a,\,b]$, then $f$ attains an absolute maximum value and an absolute minimum value at some points in $[a,\,b]$. 'Attains' is the key word — the function does not merely approach these values, it actually reaches them. Both hypotheses matter. If the interval is not closed, extremes can escape: on the OPEN interval $(0,\,1)$ the function $f(x)=x$ has no maximum and no minimum, because it can get arbitrarily close to $0$ and $1$ without ever touching them. If the function is not continuous, a jump can let a value be approached but never reached. Closed interval plus continuity together rule out both escape routes.