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Connecting a Function
and Its Derivatives

Understand how \( f \), \( f' \), and \( f'' \) are linked: how the first derivative reveals increasing/decreasing behavior and how the second derivative reveals concavity.

AP Calculus AB

What Does \( f'(x) \) Tell Us?

The first derivative \( f'(x) \) represents the instantaneous rate of change of \( f(x) \). Its sign tells us whether the original function is rising or falling at any point.

\( f'(x) > 0 \)
\( f \) is increasing on that interval
\( f'(x) < 0 \)
\( f \) is decreasing on that interval
\( f'(x) = 0 \)
Candidate for a local extremum (critical point)

A few readings leave the intervals unseen

A probe touches a plain metal object and connects to a blank logger beside a notebook and separated blank reading cards.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

A few readings leave the intervals unseen

Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.Separated measurements leave the behavior between samples undetermined. Two differentiable curves can pass through the same readings while having different derivatives between them.

What Does \( f''(x) \) Tell Us?

The second derivative \( f''(x) \) is the rate of change of the first derivative. It describes the concavity of \( f \): whether the curve bends upward or downward.

\( f''(x) > 0 \)
\( f \) is concave up (cup shape, slopes increasing)
\( f''(x) < 0 \)
\( f \) is concave down (cap shape, slopes decreasing)
\( f''(x) = 0 \)
Candidate for an inflection point (concavity change)

A continuous route through hills: elevation, grade, and intermediate height

A walking path crosses a ridge and hollow beside a staff-bearing observer.For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.

Height, grade and changing grade

For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.

Seeing \( f \), \( f' \), and \( f'' \) Together

x f(x) f'(x) f''(x) local max local min inflection

Critical Points and the First Derivative Test

A critical point occurs where \( f'(c) = 0 \) or \( f'(c) \) is undefined. To classify it, check how the sign of \( f' \) changes around \( c \):

1
If \( f' \) changes from positive to negative at \( c \), then \( f(c) \) is a local maximum.
2
If \( f' \) changes from negative to positive at \( c \), then \( f(c) \) is a local minimum.
3
If \( f' \) does not change sign, then \( c \) is neither a max nor a min.

The Second Derivative Test

An alternative way to classify critical points: evaluate \( f'' \) at the critical point \( c \) where \( f'(c) = 0 \).

\( f''(c) > 0 \)
\( f \) is concave up at \( c \), so \( f(c) \) is a local minimum
\( f''(c) < 0 \)
\( f \) is concave down at \( c \), so \( f(c) \) is a local maximum

First vs. Second Derivative Test

First Derivative Test Second Derivative Test
What you check Sign change of \( f' \) around \( c \) Value of \( f''(c) \)
Requires \( f'(c)=0 \) or undefined \( f'(c)=0 \) and \( f'' \) exists
Always conclusive? Yes No (fails when \( f''(c)=0 \))
Best when \( f'' \) is hard to compute \( f'' \) is easy to evaluate

Inflection Points and Concavity Changes

An inflection point is where the concavity of \( f \) switches. This occurs where \( f''(x) = 0 \) or is undefined, and \( f'' \) changes sign.

Inflection point Concave down \( f'' < 0 \) Concave up \( f'' > 0 \)

How to Analyze a Function Completely

1
Find \( f'(x) \). Differentiate to get the first derivative.
2
Find critical points. Solve \( f'(x) = 0 \) and note where \( f' \) is undefined.
3
Sign chart for \( f' \). Determine where \( f \) is increasing or decreasing.
4
Find \( f''(x) \). Differentiate again to get the second derivative.
5
Sign chart for \( f'' \). Determine where \( f \) is concave up or concave down.
6
Classify. Use the derivative tests to identify local extrema and inflection points.

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

1
\( f'(x) = 3x^2 - 3 = 3(x-1)(x+1) \)
2
Critical points: \( f'(x)=0 \) at \( x = -1 \) and \( x = 1 \).
3
\( f''(x) = 6x \). At \( x=-1 \): \( f''(-1) = -6 \) \( < 0 \) → local max. At \( x=1 \): \( f''(1) = 6 \) \( > 0 \) → local min.

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

Step 1: \( g'(x) = 4x^3 - 12x^2 = 4x^2(x - 3) \). Critical points at \( x = 0 \) and \( x = 3 \).

Step 2: \( g''(x) = 12x^2 - 24x = 12x(x - 2) \). Evaluate at critical points:

\( x = 0 \)
\( g''(0) = 0 \): test is inconclusive. Use the first derivative test: \( g' \) does not change sign (negative on both sides), so neither max nor min.
\( x = 3 \)
\( g''(3) = 36 > 0 \): concave up, so \( x = 3 \) is a local minimum with \( g(3) = -27 \).

Example 2 continued: Inflection Points

\( g''(x) = 12x(x-2) = 0 \) at \( x = 0 \) and \( x = 2 \).

Interval Sign of \( g'' \) Concavity of \( g \)
\( x < 0 \) \( + \) Concave up
\( 0 < x < 2 \) \( - \) Concave down
\( x > 2 \) \( + \) Concave up

Example 3: Reading the Graph of \( f' \)

Given the graph of \( f'(x) \) below, determine where \( f \) has local extrema and inflection points.

x \( f' \) a b c \( f' = 0 \) \( f' = 0 \) \( f' > 0 \) \( f' < 0 \) \( f' > 0 \)

The Chain of Information

\( f(x) \)
 differentiate
\( f'(x) \)
 differentiate
\( f''(x) \)
 Position / Value Slope / Rate Concavity / Acceleration

Common Mistakes to Avoid

✕ Assuming \( f'(c)=0 \) means extremum
\( f'(c) = 0 \) only gives a candidate. You must verify with a sign change test. Example: \( f(x) = x^3 \) has \( f'(0)=0 \) but no extremum.
✕ Confusing \( f' = 0 \) with \( f'' = 0 \)
\( f'=0 \) → critical point (possible extremum). \( f''=0 \) → possible inflection point. These are different features of \( f \).
✕ Forgetting to check sign change for inflection
\( f''(c) = 0 \) is necessary but not sufficient. Example: \( f(x)=x^4 \) has \( f''(0)=0 \) but no inflection point because concavity doesn't change.

Summary: The Complete Picture

Condition What it means for \( f \)
\( f'(x) > 0 \) \( f \) is increasing
\( f'(x) < 0 \) \( f \) is decreasing
\( f'(c) = 0 \), sign change \( + \to - \) Local maximum at \( x = c \)
\( f'(c) = 0 \), sign change \( - \to + \) Local minimum at \( x = c \)
\( f''(x) > 0 \) \( f \) is concave up
\( f''(x) < 0 \) \( f \) is concave down
\( f''(c) = 0 \), sign change in \( f'' \) Inflection point at \( x = c \)

Question 1 of 5

If \( f'(x) > 0 \) for all \( x \) in the interval \( (a, b) \), which of the following must be true?

Question 2 of 5

Let \( f(x) = x^4 - 4x^2 \). At \( x = 0 \), the second derivative test gives \( f''(0) = -8 \). What can you conclude?

Question 3 of 5

The graph of \( f' \) changes from negative to positive at \( x = 3 \). What is true about \( f \) at \( x = 3 \)?

Question 4 of 5

For \( h(x) = x^3 \), we find \( h''(0) = 0 \). Does \( h \) have an inflection point at \( x = 0 \)?

Question 5 of 5

If \( f''(x) > 0 \) for all \( x \) in \( (a, b) \), and \( f'(c) = 0 \) for some \( c \) in \( (a, b) \), then \( f(c) \) is: