Interpreting the Graph of f'

Using the graph of the derivative to determine features of the original function $f(x)$.

Core Theorem

First Derivative Test:

f

has a relative max where

f'

changes from positive to negative. Concavity:

f

is concave up where

f'

is increasing (

f'' > 0

Practice Exercises

Example 01Easy

Given the graph of

f'

, if

f'

crosses from above the x-axis to below at

x=3

, what happens to

f

x=3

NEED A HINT?

Check the sign change of

f'

SHOW DETAILED EXPLANATION

Conclusion

f(x)

has a relative maximum at

x=3

because

f'

changes from positive to negative.

Gary Chang

Calculus Educator

5+ Years of Calculus Teaching Experience | AP Calculus Specialist. Dedicated to helping students master calculus through step-by-step logic and clear visualizations.

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