We say that a function *f* is **increasing** on an interval (*a, b*) if *f(x _{2}) > f(x_{1})* whenever

For the interval (*a, b*), if *f' > 0*, then *f* is increasing, and if *f' < 0*, then *f* is decreasing.

A real number *x* in the domain of *f* such that f'(x) = 0 or *f'(x)* does not exist is called a **critical number** of *f*.

*f(c)*is a**local maximum**if there exists an interval (*m, n*) containing*c*such that*f(x) ≤ f(c)*for all*x*in (*m, n*).*f(c)*is a**local minimum**if there exists an interval (*m, n*) containing*c*such that*f(x) ≥ f(c)*for all*x*in (*m, n*).- If
*f(c)*is either a local maximum or a local minimum, it is called a**local extremum**.

If *f(c)* is a local extremum of the function *f*, then *c* is a critical number of *f*.

Let *c* be a critical number of *f* [*f(c)* is defined and either *f'(c) = 0* or *f'(c)* is not defined]. Construct a sign chart for *f'(x)* close to and on either side of *c*.

- If
*f'(x)*changes from negative to positive at*c*, then*f(c)*is a local minimum. - If
*f'(x)*changes from positive to negative at*c*, then*f(c)*is a local maximum. - If
*f'(x)*does not change sign at*c*, then*f(c)*is neither a local maximum nor a local minimum.

If:

f(x) = a_{n}x^{n}+ a_{n-1}x^{n-1}+ ... + a_{1}x + a_{0}, a_{n}≠ 0

is a polynomial function of degree *n ≥ 1*, then *f* has at most * n x* intercepts and at most *n - 1* local extrema.