How do I find where f has a relative maximum from the graph of f prime?

Find where the graph of f prime crosses the x-axis from above to below. That's where f prime changes from positive to negative.

How do I see concavity on the graph of f prime?

Look at whether the graph of f prime is going up or down. Going up means f is concave up, and going down means f is concave down.

Are inflection points where f prime is zero?

Not necessarily. Inflection points of f are where f prime switches between increasing and decreasing, which happens at a high or low point of the graph of f prime.

Can I find values of f from the graph of f prime?

Yes, if you know one value of f. Add the signed area under the graph of f prime as you move right from that point, and subtract it if you move left.

Reading the graph of a derivative for AP Calculus AB

Updated November 9, 2026

A classic AP Calculus AB question shows you the graph of $f'$ and asks about $f$ . The trap is reading it as if it were the graph of $f$ . Its height is the slope of $f$ , not the value of $f$ .

This guide shows what to look for on the graph of $f'$ , how to write each justification, and how to find actual values of $f$ with area.

What the graph of f' tells you

$f$ is increasing where the graph of $f'$ is above the $x$ -axis.
$f$ is decreasing where the graph of $f'$ is below the $x$ -axis.
$f$ has a relative minimum where $f'$ changes from negative to positive.
$f$ has a relative maximum where $f'$ changes from positive to negative.
$f$ is concave up where the graph of $f'$ is increasing.
$f$ is concave down where the graph of $f'$ is decreasing.
$f$ has an inflection point where $f'$ switches between increasing and decreasing, as long as $f$ is continuous there.

Height above or below the axis answers increasing and decreasing. The direction the graph of $f'$ is moving answers concavity. These readings assume $f$ is continuous, which is automatic wherever $f'$ exists.

A worked example

y = f'(x)

The graph of f prime, made of straight segments. It touches the axis at x = 5 without crossing.

Five questions from one graph

The graph above is the graph of

f'

, the derivative of a function

f

, on

[0, 10]

, and

f(0) = 5

. On the open interval

(0, 10)

: (a) Where is

f

increasing? (b) Where does

f

have relative extrema? (c) Where is

f

concave down? (d) Where does

f

have inflection points? (e) Find the absolute maximum and minimum values of

f

[0, 10]

(a) $f' > 0$ on $(1, 5)$ and on $(5, 7)$ , so $f$ is increasing on those intervals. At $x = 5$ , $f'$ only touches zero, so $f$ keeps increasing through it, and you can say $f$ is increasing on $[1, 7]$ .

(b) $f$ has a relative minimum at $x = 1$ , because $f'$ changes from negative to positive there. $f$ has a relative maximum at $x = 7$ , because $f'$ changes from positive to negative there. There's no extremum at $x = 5$ , because $f'$ doesn't change sign.

(d) $f'$ switches between increasing and decreasing at $x = 3$ , $x = 5$ , $x = 6$ , and $x = 8$ , so those are the inflection points.

(e) Candidates are the endpoints and the points where $f'$ changes sign. Use signed area, starting from $f(0) = 5$ . The areas are $-1.5$ on $[0, 1]$ , $2 + 2 + 1 + 1 = 6$ on $[1, 7]$ , and $-3$ on $[7, 10]$ .

$f(0) = 5$
$f(1) = 5 - 1.5 = 3.5$
$f(7) = 3.5 + 6 = 9.5$
$f(10) = 9.5 - 3 = 6.5$

f'

f

The function and its derivative together. Where the derivative is below the axis, the function falls.

Answer: (a) $[1, 7]$ . (b) Relative minimum at $x = 1$ , relative maximum at $x = 7$ . (c) $(3, 5)$ and $(6, 8)$ . (d) $x = 3, 5, 6, 8$ . (e) Maximum $9.5$ at $x = 7$ , minimum $3.5$ at $x = 1$ .

How to write the justification

Name $f'$ , describe what it does, and say where. This is the kind of reasoning scoring guidelines look for.

Increasing: " $f$ is increasing on $(1, 5)$ because $f'(x) > 0$ on that interval."
Relative maximum: " $f$ has a relative maximum at $x = 7$ because $f'$ changes from positive to negative at $x = 7$ ."
Concavity: "The graph of $f$ is concave down on $(3, 5)$ because $f'$ is decreasing on that interval."
Inflection: "The graph of $f$ has an inflection point at $x = 3$ because $f'$ changes from increasing to decreasing at $x = 3$ ."
Absolute extremum: list the values of $f$ at the candidates, then compare them.

More wording for other question types is in the free-response guide.

The same logic from a formula

A derivative given in factored form

Let

f'(x) = (x - 2)^2(x + 1)

. Find the

x

-values of all relative extrema of

f

and the

x

-values of all inflection points of the graph of

f

$f'$ is zero at $x = -1$ and $x = 2$ . The factor $(x - 2)^2$ is never negative, so the sign of $f'$ is the sign of $x + 1$ everywhere else.

$f'$ changes from negative to positive at $x = -1$ , so that's a relative minimum. $f'$ is positive on both sides of $x = 2$ , so there's no extremum at $x = 2$ .

Using the product rule and factoring, $f''(x) = 2(x - 2)(x + 1) + (x - 2)^2 = 3x(x - 2)$ . It changes sign at $x = 0$ and at $x = 2$ , so both are inflection points.

Answer: Relative minimum at $x = -1$ , inflection points at $x = 0$ and $x = 2$

A quick check

Which description fits

On the interval

(2, 5)

, the graph of

f'

is above the

x

-axis and decreasing. Which statement describes

f

(2, 5)

A. $f$ is increasing and concave up
B. $f$ is increasing and concave down
C. $f$ is decreasing and concave up
D. $f$ is decreasing and concave down

Above the axis means $f' > 0$ , so $f$ is increasing. The graph of $f'$ is going down, so $f$ is concave down.

Answer: B

When the function is defined by an integral, the same reading applies. See the fundamental theorem and accumulation guide.

Keep going

Practice on exam-style questions

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Questions

How do I find where f has a relative maximum from the graph of f prime?: Find where the graph of f prime crosses the x-axis from above to below. That's where f prime changes from positive to negative.
How do I see concavity on the graph of f prime?: Look at whether the graph of f prime is going up or down. Going up means f is concave up, and going down means f is concave down.
Are inflection points where f prime is zero?: Not necessarily. Inflection points of f are where f prime switches between increasing and decreasing, which happens at a high or low point of the graph of f prime.
Can I find values of f from the graph of f prime?: Yes, if you know one value of f. Add the signed area under the graph of f prime as you move right from that point, and subtract it if you move left.