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First Derivative
Test

Master the technique of using \( f'(x) \) sign changes to identify relative maxima and relative minima of a function.

AP Calculus AB

What Are Relative Extrema?

A function \( f \) has a relative (local) maximum at \( x = c \) if \( f(c) \geq f(x) \) for all \( x \) near \( c \). Similarly, \( f \) has a relative (local) minimum at \( x = c \) if \( f(c) \leq f(x) \) for all \( x \) near \( c \).

Relative Max
The highest point in a neighborhood: the function rises to \( c \) then falls.
Relative Min
The lowest point in a neighborhood: the function falls to \( c \) then rises.

Critical Numbers: Where Extrema Live

A number \( c \) in the domain of \( f \) is a critical number if \( f'(c) = 0 \) or \( f'(c) \) is undefined. Relative extrema can only occur at critical numbers.

\( f'(c) = 0 \)
The tangent line is horizontal: the curve flattens out momentarily.
\( f'(c) \) DNE
The derivative doesn't exist: a cusp, corner, or vertical tangent occurs.

The Sign of \( f'(x) \) Tells the Story

Rel. Max Rel. Min
\( f'(x) > 0 \)
 Increasing
\( f'(x) < 0 \)
 Decreasing
\( f'(x) > 0 \)
 Increasing

The First Derivative Test

Let \( c \) be a critical number of a continuous function \( f \).

1
If \( f' \) changes from positive to negative at \( c \), then \( f \) has a relative maximum at \( c \).
2
If \( f' \) changes from negative to positive at \( c \), then \( f \) has a relative minimum at \( c \).
3
If \( f' \) does not change sign at \( c \), then \( f \) has no relative extremum at \( c \).

Sign Change Patterns

Relative Maximum + + + c - - - Relative Minimum - - - c + + + No Extremum + + + c + + +

Relative vs Absolute Extrema

Relative (Local) Absolute (Global)
Definition Largest/smallest in a neighborhood Largest/smallest on the entire domain
Location Only at critical numbers At critical numbers or endpoints
Method First Derivative Test Closed Interval Method (EVT)
Count Can have many At most one max, one min

How to Apply the Test

1
Find \( f'(x) \). Differentiate the function.
2
Find critical numbers. Solve \( f'(x) = 0 \) and identify where \( f'(x) \) is undefined.
3
Build a sign chart. Pick test points in each interval between critical numbers and evaluate the sign of \( f'(x) \).
4
Read the sign changes. \( + \to - \): relative max. \( - \to + \): relative min. No change: no extremum.
5
Find the \( y \)-value. Substitute each critical number back into \( f(x) \) for the extremum coordinates.

Example 1: \( f(x) = x^3 - 3x + 2 \)

Step 1: \( f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1) \)

Step 2: Critical numbers: \( f'(x) = 0 \) when \( x = -1 \) and \( x = 1 \).

-1 1 + − +

Example 1: Graph

Example 2: \( g(x) = x^4 - 4x^3 \)

Step 1: \( g'(x) = 4x^3 - 12x^2 = 4x^2(x - 3) \)

Step 2: Critical numbers: \( g'(x) = 0 \) when \( x = 0 \) and \( x = 3 \).

0 3 − − +

Example 3: \( h(x) = x^{2/3} \)

Step 1: \( h'(x) = \frac{2}{3}x^{-1/3} = \frac{2}{3x^{1/3}} \)

Step 2: \( h'(x) \) is never zero, but \( h'(0) \) is undefined. So \( x = 0 \) is the only critical number.

0 − +

Why Does This Work?

The First Derivative Test is built on a fundamental connection between the derivative and the monotonicity of a function:

1
\( f'(x) > 0 \) on an interval implies \( f \) is strictly increasing there (by the Mean Value Theorem).
2
\( f'(x) < 0 \) on an interval implies \( f \) is strictly decreasing there.
3
A switch from increasing to decreasing must create a peak, and a switch from decreasing to increasing must create a valley.

Common Mistakes

✕ Every \( f'(c) = 0 \) is an extremum
\( f(x) = x^3 \) has \( f'(0) = 0 \) but no sign change at 0: no extremum there.
✕ Forgetting \( f' \) DNE points
Critical numbers include where \( f' \) is undefined. The cusp in \(|x|\) gives a minimum at \(x=0\).
✕ Wrong test-point intervals
You must test one point in each interval created by the critical numbers. Skipping an interval can miss a sign change.

First vs Second Derivative Test

First Derivative Test Second Derivative Test
Uses Sign of \( f'(x) \) near \( c \) Value of \( f''(c) \)
Works when \( f'(c) \) DNE? Yes No
Inconclusive case Never (always gives an answer) When \( f''(c) = 0 \)
Extra info Increasing/decreasing intervals Concavity at the point
Effort More work (sign chart) Quick (one evaluation)

Question 1 of 5

If \( f'(x) \) changes from positive to negative at \( x = c \), what does the First Derivative Test conclude?

Question 2 of 5

Let \( f(x) = x^3 - 12x + 5 \). At which value(s) of \( x \) does \( f \) have a relative minimum?

Question 3 of 5

If \( f'(x) = x^2(x - 5) \), what does the First Derivative Test tell us about \( x = 0 \)?

Question 4 of 5

A continuous function \( f \) has \( f'(3) \) undefined. For \( x < 3 \), \( f'(x) < 0 \); for \( x > 3 \), \( f'(x) > 0 \). What is \( x = 3 \)?

Question 5 of 5

Which statement about the First Derivative Test is true?