Section 3.3 Absolute Extrema
Consider an interval in the domain of a function \(f\) and a point \(x_0\text{.}\) \(f\) has an absolute maximum at \(x_0\) if \(f(x) \leq f(x_0)\) for all \(x\) in the interval. \(f\) has an absolute minimum at \(x_0\) if \(f(x) \geq f(x_0)\) for all \(x\) in the interval. \(f\) has an absolute extrema at \(x_0\) if \(f\) has either of the above at that point. Absolute minimum on \(( - \infty, \infty)\text{,}\) no absolute maximum.
Definition 3.3.1.
Example 3.3.1.
No exrema on \(( - \infty, \infty)\text{.}\)
Absolute maximum and absolute minimum on \((- \infty, \infty)\text{.}\)
Theorem 3.3.5. Absolute Value Theorem.
If a function \(f\) is continuous on a finite, closed interval, \([a,b]\text{,}\) then \(f\) has both an absolute minimum and absolute maximum on \([a,b]\text{.}\)
Extreme values will occur at critical points in the interval or at the end points.
Method to find Absolute Extrema:
find critical points
evaluate \(f\) at critical points and endpoints
compare function values and decide
Example 3.3.2.
Find the absolute extrema of \(f(x) = 2x^6 - 4x^3\) on the interval \([-1,1]\text{:}\)
(\(f\) has critical points at \(x = 0,1\))
| Type | \(x\) | \(f(x)\) | Result |
| EP | \(-1 \) | \(6 \) | absolute maximum on \([-1,1]\) |
| CP | \(0\) | \(0\) | ----- |
| EP/CP | \(1\) | \(-2\) | absolute minimum on \([-1,1]\) |
Note 3.3.6.
Changing the interval is likely to change the answer.