Maxima and Minima of Single Variable Functions

1. Understanding Maxima and Minima

Maxima and minima refer to the highest and lowest points of a function, respectively. These points can be classified as local (relative) or global (absolute).

• Local Maximum: A point where the function has a higher value than at nearby points.
• Local Minimum: A point where the function has a lower value than at nearby points.
• Global Maximum/Minimum: The highest/lowest value of the function on its entire domain.

2. First Derivative Test

The first derivative test involves examining the sign changes of the derivative \( f'(x) \) to determine local maxima or minima.

• If \( f'(x) \) changes from positive to negative at \( x = c \), then \( f(c) \) is a local maximum.
• If \( f'(x) \) changes from negative to positive at \( x = c \), then \( f(c) \) is a local minimum.
• If no sign change occurs, the point is neither a maximum nor a minimum.

3. Second Derivative Test

The second derivative test is used to confirm the nature of critical points found using the first derivative.

• If \( f''(x) > 0 \), the function is concave up and the point is a local minimum.
• If \( f''(x) < 0 \), the function is concave down and the point is a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive.

4. Example

Find the local maxima and minima of \( f(x) = x^3 - 6x^2 + 9x + 15 \).

1. First derivative: \( f'(x) = 3x^2 - 12x + 9 \)
2. Set \( f'(x) = 0 \): \( 3x^2 - 12x + 9 = 0 \Rightarrow x = 1, 3 \)
3. Second derivative: \( f''(x) = 6x - 12 \)
4. At \( x = 1 \), \( f''(1) = -6 \) → Local Max at \( x = 1 \)
5. At \( x = 3 \), \( f''(3) = 6 \) → Local Min at \( x = 3 \)