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Section 4.1 Extreme Values of Functions

Definition 4.1.1. Global Extrema.

Let \(f\) be a function with domain \(D\) (an interval or union of intervals). Then \(f\) has an absolute maximum value on \(D\) at a point \(c\) if \(f(c) \geq f(x)\) for all \(x\) in \(D\text{.}\) And \(f\) has an absolute minimum value on \(D\) if \(f(x) \leq f(x)\) for all \(x\) in \(D\text{.}\)

Example 4.1.2.

Find the absolute maxima and minima for the functions displayed below.

Graph of parabola on half-closed domain
Figure 4.1.3. \(f(x)=x^2\text{,}\) \([-2,\infty)\)
Graph of cubic on half-open half-closed domain
Figure 4.1.4. \(g(x)=x^3\text{,}\) \((-2,2]\)
Graph of line on open interval
Figure 4.1.5. \(h(x)=x\text{,}\) \((-2,2)\)
Graph of sine
Figure 4.1.6. \(k(x)=\sin(x)\text{,}\) \((-\infty,\infty)\)

Note: There may be more than one absolute maximum and more than one absolute minimum.

Definition 4.1.8. Relative extrema.

A function \(f\) has a local maximum value at a point \(c\) in its domain \(D\) if \(f(c)\geq f(x)\) for all \(x\in D\) within some open interval around \(c\text{.}\)

A function \(f\) has a local minimum value at a point \(c\) in its domain \(D\) if \(f(c)\leq f(x)\) for all \(x\in D\) within some open interval around \(c\text{.}\)

Example 4.1.9.

The function graphed below has:

  • A local minimum of \(1.88\) at \(x=-2.5\)

  • A local maximum of \(3.35\) at \(x=2\)

  • An absolute minimum of \(-0.71\) at \(x=0.64\)

  • An absolute maximum of \(4.8\) at \(x=2\)

Graph with extrema
Figure 4.1.10. Extrema

Definition 4.1.12. Critical point.

An interior point \(c\) of the domain of \(f\) is a critical point if either:

  • \(f'(c)=0\text{,}\) or

  • \(f'\) is undefined at \(c\text{,}\) but \(f\) is defined at \(c\text{.}\)

Example 4.1.13.

  1. Find the critical points of the function \(\displaystyle f(x) = \sqrt{-x^2+4}\text{.}\)

  2. Find the critical points of the function \(\displaystyle g(x) = \frac{\ln x}{x}\text{.}\)

Answer.

  1. \(\displaystyle -2,0,2\)

  2. \(\displaystyle e\)

Example 4.1.15.

  1. Find the absolute extrema of \(f(x) = |x|\) on the interval \(-2 \leq x \leq 4\text{.}\)

  2. Find the absolute extrema of \(g(x) = \frac{\ln x}{x}\) on the interval \([1,5]\text{.}\)

  3. Find the absolute extrema of \(h(x) = \frac{1}{10}(x-1)^2(x+3)^3\) on the interval \([-4,1\frac{1}{2}]\text{.}\)

  4. Find the absolute extrema of \(k(x) = 3 \cos^{-1}(x^2)\text{.}\)

Answer.

  1. abs min \((0,0)\text{;}\) abs max \((4,4)\)

  2. \(g(1)=0\text{,}\) \(g(e) \approx 0.3679\text{,}\) \(g(5) \approx 0.3219\)

    abs min \((1,0)\text{;}\) abs max \((e,\frac{1}{e}\)

  3. \(h(-4)=-2.5\text{,}\) \(h(-3)=0\text{,}\) \(h(-3/5)\approx 3.5389\text{,}\) \(h(3/2)\approx 2.2781\)

    abs min \((-4,-2.5)\text{;}\) abs max at \(x=-3/5\)

  4. abs min \((1,0)\) and \((-1,0)\text{;}\) abs max \((-1,3\pi)\)

Note: results about extreme values can be written as either:

  • The absolute maximum value \(5\) occurs at \(x=2\text{;}\) or

  • The absolute maximum occurs at \((2,5)\text{.}\)