Second Derivative

The Second Derivative

The second derivative $f''(x)$ is the derivative of $f'(x)$ — the rate of change of the slope.

Geometric Meaning

• $f''(x) > 0$: curve is concave up (holds water, like a bowl)
• $f''(x) < 0$: curve is concave down (spills water, like a hill)
• $f''(x) = 0$: possible inflection point (concavity changes)

Numerical Formula

The second derivative can be approximated directly from $f$ without computing $f'$ first:

$f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$

Derivation: expand $f(x+h)$ and $f(x-h)$ using Taylor series:

$f(x+h) = f(x) + h f'(x) + \frac{h^2}{2} f''(x) + O(h^3)$

$f(x-h) = f(x) - h f'(x) + \frac{h^2}{2} f''(x) + O(h^3)$

Add them: $f(x+h) + f(x-h) = 2f(x) + h^2 f''(x)$, so:

$f''(x) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$

Inflection Points

At an inflection point, $f''(c) = 0$ and the sign of $f''$ changes. Example: $f(x) = x^3$ has an inflection at $x = 0$.

Second Derivative Test

For a critical point $c$ where $f'(c) = 0$:

• $f''(c) > 0$ → local minimum
• $f''(c) < 0$ → local maximum
• $f''(c) = 0$ → inconclusive

Your Task

Implement double second_derivative(double (*f)(double), double x, double h).