The second derivative is the derivative of the derivative. It reports how the slope is changing.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Concavity
Positive means the slope is increasing and the curve holds water, called concave up. Negative means concave down.
Inflection points
An inflection point is where concavity changes. The second derivative must change sign, not merely equal zero.
The Second Derivative Test
At a critical point, a negative second derivative means a maximum and a positive one means a minimum.
Why that works
A flat point on a concave-down curve is a peak. There is nowhere to go but down on both sides.
When it fails
If the second derivative is zero the test says nothing, and the first derivative sign chart must be used instead.
How the slope itself is changing
The second derivative measures the rate of change of the slope. Positive means the slope is increasing and the curve bends upwards; negative means it bends downwards. It describes curvature rather than direction.
Inflection points
An inflection point is where concavity changes. It requires f″ to change sign, not merely to be zero — f(x) = x⁴ has f″ zero at the origin with no change in concavity, so it is not an inflection point.
The second derivative test
At a critical point, a positive f″ means a local minimum and a negative f″ a local maximum. If f″ is zero the test is inconclusive and the first derivative test must be used instead.
Concavity carries real meaning
During an epidemic, a positive first derivative means cases are rising; a negative second derivative means they are rising more slowly. The two together are what "flattening the curve" actually described.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Slope of the tangent-3
Step 3: Watch an Example
One step at a time.
Watch Amara Classify a Critical Point
Amara has f(x) = x³ − 3x and needs to classify the critical point at x = 1.
- Step 1
The first derivative is 3x² − 3, and setting it to zero gives x = 1 and x = −1.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
f′(2) = 0 and f″(2) = −5. Is x = 2 a maximum or minimum? 1 max, 2 min.
The Inflection
Problem 2 of 2
f″(x) = 6x. At which x is there an inflection point?
Read the Bend
1 of 8
f″(3) = 4. Concave up or down? 1 up, 2 down.
2 of 8
f′(5) = 0 and f″(5) = 2. Max or min? 1 max, 2 min.
3 of 8
f(x) = x², so f″(x) = 2. Concave up or down everywhere? 1 up, 2 down.
4 of 8
f(x) = x³ − 3x, so f″(x) = 6x. What is f″(−2)?
5 of 8
f′(1) = 0 and f″(1) = 0. Is the test conclusive? 1 for yes, 0 for no.
6 of 8
How many inflection points does a cubic have?
7 of 8
Sort each sign of the second derivative by what it means.
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- Empty
- Empty
8 of 8
f(x) = x⁴, so f″(x) = 12x². What is f″(2)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f′(4) = 0 and f″(4) = −3. Max or min? 1 max, 2 min.
Question 2 of 2
What does an inflection point require?
What You Learned
- The second derivative reports how the slope is changing.
- Positive means concave up, negative means concave down.
- At a critical point, the sign of the second derivative names a maximum or a minimum.