Maxima and Minima

Maxima and minima explained using critical points, first derivative test, second derivative test, and simple examples.

1. What Maxima and Minima Mean

Maxima and minima are points where a function reaches a highest or lowest value in a region. In notes-style: maxima are peaks, minima are valleys. These points help understand the overall shape and behaviour of a function.

2. Local vs Global Extrema

A local maximum is a point where the function is higher than nearby values. A local minimum is a point where the function is lower than nearby values.

A global maximum or minimum is the absolute highest or lowest point on the entire domain.

3. Critical Points

Critical points are where the function may achieve maxima or minima. A critical point occurs when:

  • f'(x) = 0, or
  • f'(x) is undefined

These points divide the graph into intervals that can be tested for increasing or decreasing behaviour.

4. First Derivative Test

The first derivative test checks how the sign of f'(x) changes around a critical point.

4.1. Rules

  • If f' changes from + to −, the point is a local maximum.
  • If f' changes from − to +, the point is a local minimum.
  • If f' does not change sign, no extremum occurs.

4.2. Example

f(x) = x^3 - 3x

Derivative:

f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1)

Critical points: x = -1, x = 1

  • At x = -1: derivative changes from + to − → local maximum
  • At x = 1: derivative changes from − to + → local minimum

5. Second Derivative Test

The second derivative test analyses the curvature at a critical point.

5.1. Rules

  • If f'(a) = 0 and f''(a) > 0 → local minimum
  • If f'(a) = 0 and f''(a) < 0 → local maximum
  • If f'(a) = 0 and f''(a) = 0 → test is inconclusive

5.2. Example

f(x) = x^2

f'(x) = 2x, f''(x) = 2

At x = 0, f'(0) = 0 and f''(0) = 2 > 0 → local minimum.

6. Identifying Extrema — Step-by-Step

  1. Find the derivative f'(x).
  2. Solve f'(x) = 0 for critical points.
  3. Use the first or second derivative test.
  4. Identify whether each point is a maximum, minimum, or neither.

7. Graphical Understanding

The slope of the tangent line gives clear hints about maxima and minima:

  • At maxima: slope goes from positive to negative.
  • At minima: slope goes from negative to positive.
  • At flat points with no sign change: no extremum occurs.

8. More Examples

These examples strengthen the idea.

8.1. Example 1 — Polynomial

f(x) = x^4 - 4x^2

f'(x) = 4x^3 - 8x = 4x(x^2 - 2)

Critical points: x = 0, x = √2, x = −√2

Test each point using sign changes or second derivative.

8.2. Example 2 — Trigonometric

f(x) = \sin x

f'(x) = \cos x

Critical points where cos x = 0 → x = \( \frac{\pi}{2} + n\pi \)

At x = π/2 → local maximum
At x = 3π/2 → local minimum

9. Why Maxima and Minima Matter

Maxima and minima help understand important turning points of a function. They play a key role in optimisation, graph sketching, and analysing the behaviour of curves in many real situations.