At a maximum or minimum the tangent is horizontal, so the derivative is zero there.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The first derivative test
If f′ changes from positive to negative, that is a maximum. Negative to positive is a minimum.
The second derivative test
f″ positive means the curve is concave up, so a critical point there is a minimum.
Optimisation
Write the quantity to optimise as a function of one variable, differentiate, and set it to zero.
Critical points
A critical point is where f′ is zero or undefined. Extrema can only occur at critical points or at the endpoints of the interval, which reduces the search from infinitely many points to a short list.
Not every critical point is an extremum
For f(x) = x³, the derivative is zero at the origin and there is no maximum or minimum — the graph flattens and carries on. A critical point is a candidate, not a conclusion.
The first derivative test
If f′ changes from positive to negative, the point is a local maximum; negative to positive gives a local minimum; no change gives neither. Checking the sign either side settles the classification.
Local against global
A local maximum is highest in its neighbourhood; a global maximum is highest overall. On a closed interval, compare every critical value with the endpoint values — the global extremum may sit at an endpoint.
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 Priya Maximise an Area
20 m of fence makes a rectangle. Priya finds the largest area.
- Step 1
The perimeter gives 2l + 2w = 20, so w = 10 − l.
Step 4: Your Turn
Practice makes it stick.
The Critical Point
Problem 1 of 2
f′(x) = 10 − 2x. At what x is it zero?
The Fence
Problem 2 of 2
24 m of fence makes a rectangle. What is the largest area?
square m
Find the Best
1 of 4
f′(x) = 8 − 2x. At what x is it zero?
2 of 4
40 m of fence makes a rectangle. Largest area?
3 of 4
f′(x) = 2x − 6. At what x is it zero?
4 of 4
f″ positive at a critical point. Maximum or minimum?
Step 5: Quick Check
Show what you know.
Question 1 of 2
32 m of fence makes a rectangle. Largest area?
Question 2 of 2
What is true of the derivative at a maximum or minimum?
What You Learned
- Extrema occur where the derivative is zero.
- The sign change of f′, or the sign of f″, tells you which kind.
- Optimisation writes the quantity as one function, then differentiates.