medium difficulty10 min

Maxima and Minima

A function has a local maximum at a point if its value is greater than or equal to nearby values, and a local minimum if its value is less than or equal to nearby values.

In one line: Positive second derivative opens upward, so it gives a minimum; negative second derivative opens downward, so it gives a maximum.

Practice This Concept

Understand it

Possible points of maxima or minima occur at critical points.
The first derivative test checks the sign change of f'(x) around the point.
The second derivative test uses f''(a): if f'(a) = 0 and f''(a) < 0, there is a local maximum; if f''(a) > 0, there is a local minimum.
If f''(a) = 0, the test is inconclusive.

Example

For f(x) = x² - 4x + 5, f'(x) = 2x - 4, so x = 2. Since f''(x) = 2 > 0, f has a local minimum at x = 2, and the minimum value is f(2) = 1.

Formula and rule

First derivative test: f' changes from positive to negative at x = a gives local maximum; f' changes from negative to positive gives local minimum.

Second derivative test: if f'(a) = 0 and f''(a) < 0, local maximum; if f''(a) > 0, local minimum; if f''(a) = 0, use another test.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find local extrema of f(x) = x³ - 3x² - 9x + 5. f'(x) = 3x² - 6x - 9 = 3(x - 3)(x + 1), so critical points are x = -1 and x = 3. f''(x) = 6x - 6.

1At x = -1, f''(-1) = -12 < 0, so local maximum; f(-1) = -1 - 3 + 9 + 5 = 10.
2At x = 3, f''(3) = 12 > 0, so local minimum; f(3) = 27 - 27 - 27 + 5 = -22.
3Thus local maximum value is 10 at x = -1 and local minimum value is -22 at x = 3.

Method you can reuse

Find f'(x); solve f'(x) = 0 and list critical points in the domain; use first derivative sign change or second derivative test; calculate f(a) for each confirmed extremum; state whether it is local maximum or local minimum.

Visual tip

A curve sketch or derivative sign chart is useful to show rising-falling behaviour around a critical point.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If f'(a) = 0 and f''(a) = -5, what type of extremum may occur at x = a?

Think first, then show answer

Since f''(a) < 0, f has a local maximum at x = a.

Common wrong answer

Using f''(a) > 0 as maximum and f''(a) < 0 as minimum, which reverses the second derivative test.

Don't mix this up

Assuming every solution of f'(x) = 0 is automatically a maximum or minimum.

Exam tip

After finding a critical point, always apply a test and state both the point and the value if asked.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Appears as a derivative-test problem, assertion-based reasoning, graph interpretation, or an optimisation step inside a longer question.

1
Enough for 1 mark
A function has a local maximum at a point if its value is greater than or equal to nearby values, and a local minimum if its value is less than or equal to nearby values.
2
Add for 2 marks
Example: For f(x) = x² - 4x + 5, f'(x) = 2x - 4, so x = 2. Since f''(x) = 2 > 0, f has a local minimum at x = 2, and the minimum value is f(2) = 1.
3
Add for 3 marks
Possible points of maxima or minima occur at critical points.

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