Where f prime is positive, f is increasing. Where it is negative, f is decreasing.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Zeros of the derivative
A zero of f prime is a critical point of f. It is a maximum or minimum only if the sign actually changes there.
A zero without a change
y = x³ has f prime = 0 at the origin, yet the function keeps rising. A flat point is not automatically an extremum.
The second derivative
Where f double prime is positive, f is concave up. Where it is negative, f is concave down.
Inflection points
An inflection point of f is where f double prime changes sign, which is where f prime turns.
The exam trap
A maximum of f prime is an inflection point of f, not a maximum of f. Always ask which function the graph shows.
The sign of f′ gives the direction of f
Where f′ is positive, f increases; where negative, f decreases; where f′ crosses zero, f has a local extremum. Reading f from a graph of f′ is one of the most common free-response tasks.
The slope of f′ gives the concavity of f
Where f′ is increasing, f is concave up; where f′ is decreasing, f is concave down. Extrema of f′ are inflection points of f. Two levels of reading, from one graph.
The value of f′ is not the value of f
A large f′ means f is changing fast, not that f is large. Confusing the height of the derivative graph with the height of the original is the single commonest error in these questions.
Area under f′ gives change in f
By the fundamental theorem, the integral of f′ over an interval is the net change in f. So areas on the derivative graph become heights on the original, which is how these questions are usually completed.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, -2)
- Slope of the tangent0
Step 3: Watch an Example
One step at a time.
Watch Sana Read a Derivative Graph
A graph of f prime is positive before x = 2, zero at x = 2, and negative after.
- Step 1
Positive f prime before 2 means f is increasing there.
Step 4: Your Turn
Practice makes it stick.
The Sign
Problem 1 of 2
f prime is negative on an interval. Is f increasing or decreasing? 1 increasing, 2 decreasing.
The Turning Point
Problem 2 of 2
f prime changes from positive to negative at x = 4. Maximum or minimum? 1 maximum, 2 minimum.
From f Prime to f
1 of 8
f prime is positive. Is f increasing or decreasing? 1 increasing, 2 decreasing.
2 of 8
f prime changes from negative to positive at x = 1. Maximum or minimum? 1 max, 2 min.
3 of 8
f double prime is positive. Concave up or down? 1 up, 2 down.
4 of 8
f prime is zero at x = 0 for y = x³. Is it an extremum? 1 yes, 0 no.
5 of 8
A maximum of f prime corresponds to what in f? 1 a maximum, 2 an inflection point.
6 of 8
f(x) = x³ − 3x, so f prime = 3x² − 3. How many critical points?
7 of 8
Match each fact about f prime with what it says about f.
Tap a card on the left to start.
8 of 8
f double prime is negative. Concave up or down? 1 up, 2 down.
Step 5: Quick Check
Show what you know.
Question 1 of 2
f prime changes from positive to negative at x = 3. Maximum or minimum? 1 max, 2 min.
Question 2 of 2
A graph shows f prime with a peak at x = 2. What does that say about f?
What You Learned
- The sign of f prime gives increasing or decreasing.
- A sign change of f prime marks a local extremum.
- The sign of f double prime gives concavity, and its change marks an inflection point.