Increasing, decreasing and extreme values | Calculus

The sign of the derivative

When the tangent line slopes up, the graph is rising; when it slopes down, the graph is falling.

Critical numbers and relative extrema

A relative (local) maximum or minimum can only occur at a critical number. But not every critical number is an extremum, so test it.

ExampleAnalyze $$f(x) = x^3 - 12x$$

$$f'(x) = 3x^2 - 12 = 3(x - 2)(x + 2)$$ . Critical numbers: $\pm 2$ .

	$$x < -2$$	$$-2 < x < 2$$	$$x > 2$$
$$f'(x)$$	$$+$$	$$-$$	$$+$$
$$f$$	increasing	decreasing	increasing

Relative maximum at $$x = -2$$ : $$f(-2) = 16$$ . Relative minimum at $$x = 2$$ : $$f(2) = -16$$ .

Absolute extrema on a closed interval

By the Extreme Value Theorem, a continuous function on $$[a, b]$$ has an absolute max and min. They occur either at critical numbers or at the endpoints.

Optimization

Practice

Quick check

At a critical number, $$f'$$ changes from negative to positive. There is:

Quick check

To find absolute extrema of a continuous function on $$[a, b]$$ , which values must you compare?

	$\(x < -2\)$	$\(-2 < x < 2\)$	$\(x > 2\)$
$\(f'(x)\)$	$\(+\)$	$\(-\)$	$\(+\)$
$\(f\)$	increasing	decreasing	increasing