Topic 5 of 7
Curve sketching and optimization
Step 15 · Calculus 1 topics
Use the first and second derivatives to find where a function rises and falls, where it bends, and its largest and smallest values.
Increasing, decreasing and extremes
Where f′ > 0 the graph rises, and where f′ < 0 it falls. A local maximum or minimum can only happen at a critical point, where f′ is 0 or does not exist.
Example: A sign chart
Find where f(x) = x³ − 3x is increasing, and its local extremes.
- f′(x) = 3x² − 3, which factors as 3(x − 1)(x + 1): zero at −1 and 1.
- f′ is positive for x < −1, negative between −1 and 1, and positive for x > 1.
- So f rises, falls, then rises again.
AnswerA local maximum f(−1) = 2 and a local minimum f(1) = −2
Concavity and the second derivative
f″ measures how the slope itself changes. Where f″ > 0 the graph bends upward, concave up like a cup. Where f″ < 0 it bends downward. A point where the bending changes is an inflection point.
Example: The second derivative test
Use f″ to classify the critical points of f(x) = x³ − 3x.
- f″(x) = 6x.
- f″(−1) = −6, so the graph bends down there: a local maximum.
- f″(1) = 6, so the graph bends up there: a local minimum.
AnswerMaximum at x = −1, minimum at x = 1
Example: An inflection point
Where does f(x) = x³ − 6x² + 5 change concavity?
- f″(x) = 6x − 12, which is 0 at x = 2.
- f″ is negative for x < 2 and positive for x > 2.
AnswerAt x = 2
| f′ | f″ | Shape |
|---|---|---|
| + | + | Rising, bending up |
| + | − | Rising, bending down |
| − | + | Falling, bending up |
| − | − | Falling, bending down |
Optimization
To find the largest or smallest value of a quantity, write it as a function of one variable, decide its domain, then compare the values at the critical points and the endpoints.
Example: A closed interval
Find the largest and smallest values of f(x) = x² − 4x + 1 on [0, 5].
- f′(x) = 2x − 4, which is 0 at x = 2.
- f(0) = 1, f(2) = −3 and f(5) = 6.
AnswerLargest 6 at x = 5, smallest −3 at x = 2
Example: A fence by a river
A farmer has 200 m of fence for a rectangular pen along a river, fencing only three sides. What is the largest area?
- Let x be each side at right angles to the river. The side along the river is 200 − 2x.
- A(x) = x(200 − 2x), for 0 < x < 100.
- A′(x) = 200 − 4x, zero at x = 50, giving a 50 m by 100 m pen.
Answer5,000 m²
Try it yourself
Work each one on paper first, then open the answer to check.
Find the critical point of f(x) = x² − 6x + 2.
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x = 3
Where is f(x) = x³ − 12x decreasing?
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Between x = −2 and x = 2
Is x = 0 a local maximum or minimum of f(x) = x⁴ + 1? Use the signs of f′.
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A local minimum: f′ = 4x³ goes from negative to positive
Find the inflection point of f(x) = x³ + 3x².
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x = −1
Find the largest value of f(x) = 6x − x² on [0, 10].
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9, at x = 3
Two positive numbers add to 20. What is the largest their product can be?
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100