First and Second Derivatives Theorems

Theorem 1 - Stationary Point

If function \( f \) has a relative minimum or maximum at \( x = a \), we either have \( f'(a) = 0 \) (stationary point) or \( f'(a) \) does not exist.

As an example, the graph of \( f \) and its derivative \( f' \) are shown above. At the maximum (\( x = 2 \)) and the minimum (\( x = -2 \)) of \( f \), \( f' = 0 \).

Theorem 2 - First Derivative Test

Let \( f \) be a continuous function.

• 2.a - If \( f'(a) = 0 \) or \( f'(x) \) does not exist at \( x = a \) and if \( f'(x) < 0 \) to the left of \( a \) and \( f'(x) > 0 \) to the right of \( a \), then \( f \) has a relative minimum at \( x = a \).
• 2.b - If \( f'(a) = 0 \) or \( f'(x) \) does not exist at \( x = a \) and if \( f'(x) > 0 \) to the left of \( a \) and \( f'(x) < 0 \) to the right of \( a \), then \( f \) has a relative maximum at \( x = a \).

Both 2.a and 2.b can clearly be verified using the graph in Figure 1 above.

• 2.c - If \( f'(a) \) has the same sign to the left and to the right of \( x = a \), then \( f \) does not have a minimum or maximum.

The graph in Figure 2 below shows the graph of a function \( f \) that has neither a minimum nor maximum, and its first derivative does not change sign.

Theorem 3 - First and Second Derivatives Test

Suppose that both \( f' \) and \( f'' \) exist at \( x = a \) and that \( f'(a) = 0 \) (stationary point).

• 3.a - If \( f''(a) > 0 \), \( f \) has a relative minimum at \( x = a \).
• 3.b - If \( f''(a) < 0 \), \( f \) has a relative maximum at \( x = a \).
• 3.c - If \( f''(a) = 0 \), no conclusion can be made; use Theorem 2 above.

Below is the graph of function \( f \), its first derivative \( f' \), and its second derivative \( f'' \). We can easily check that:

1. At the point (\( x = -2 \)) where \( f'(-2) = 0 \) and \( f''(-2) < 0 \), \( f \) has a relative maximum.
2. At the point (\( x = 2 \)) where \( f'(2) = 0 \) and \( f''(2) > 0 \), \( f \) has a relative minimum.