Section 4.1: First Derivatives and Graphs

Increasing and Decreasing Functions

We say that a function f is increasing on an interval (a, b) if f(x₂) > f(x₁) whenever a < x₁ < x₂ < b, and f is decreasing on (a, b) if f(x₂) < f(x₁) whenever a < x₁ < x₂ < b.

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

Critical Numbers

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.

Local Extrema

• 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.

Local Extrema and Critical Numbers

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

First Derivative Test for Local Extrema

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.

Intercepts and Local Extrema of Polynomial Functions

If:

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, 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.