A critical point is where f'(x) = 0 or f' is undefined. Every extremum is at a critical point.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The converse fails
Not every critical point is an extremum. x³ has a flat point at 0 that is neither a maximum nor a minimum.
The first derivative test
If f' changes from positive to negative, it is a maximum. Negative to positive is a minimum.
The second derivative test
Concave up at a critical point means a minimum; concave down means a maximum. It is silent when the second derivative is zero.
Critical points
A critical point is where f′ is zero or undefined. Extrema can occur only at critical points or at the endpoints of a closed interval, which reduces an infinite search to a finite list.
A critical point is a candidate, not a conclusion
f(x) = x³ has a critical point at the origin with no extremum. The exam requires a justification — a sign chart, or the second derivative test — not merely the location of the critical point.
The candidates test
On a closed interval, evaluate f at every critical point and at both endpoints, then compare. That is the complete method for absolute extrema, and skipping the endpoints is a common omission.
Justification is graded
Stating "f′ changes from positive to negative at x = 2, so f has a local maximum there" earns the mark. Stating the answer alone does not, however correct it is.
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 Nadia Classify Two Points
For f(x) = x³ − 3x, Nadia finds and classifies the critical points.
- Step 1
f'(x) = 3x² − 3, which is zero at x = 1 and x = −1.
Step 4: Your Turn
Practice makes it stick.
The Flat Tangent
Problem 1 of 2
f'(x) = 2x − 8. At what x is the critical point?
The Test
Problem 2 of 2
At a critical point, f'' = −5. Maximum or minimum? 1 for max, 2 for min.
Find and Classify
1 of 8
f'(x) = 2x − 10. Critical point at which x?
2 of 8
f'' = 3 at a critical point. 1 for max, 2 for min.
3 of 8
f'(x) = 3x². Critical point at which x?
4 of 8
Is that critical point of x³ an extremum? 1 for yes, 0 for no.
5 of 8
Sort each first-derivative behaviour by what it means at the critical point.
Tap something to move it.
- Empty
- Empty
- Empty
6 of 8
f'(x) = x² − 9. How many critical points?
7 of 8
f'' = 0 at a critical point. Does the second derivative test decide it? 1 for yes, 0 for no.
8 of 8
f'(x) = 4x + 12. Critical point at which x?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f'(x) = 2x + 14. Critical point at which x?
What You Learned
- Critical points are where f' is zero or undefined.
- Every extremum sits at a critical point, but not every critical point is an extremum.
- The first test watches for a sign change; the second reads concavity and is silent at zero.