Critical Points

Critical Points

A critical point is where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points are candidates for local maxima and minima.

Finding Critical Points Numerically

We use bisection on $f'(x)$: if $f'(a) < 0$ and $f'(b) > 0$ (or vice versa), by the Intermediate Value Theorem there must be a zero of $f'$ between $a$ and $b$.

$\text{while } (b - a) > \text{tolerance}:$ $\quad m = \frac{a + b}{2}$ $\quad \text{if } f'(a) \text{ and } f'(m) \text{ have the same sign: } a = m$ $\quad \text{else: } b = m$ $\text{return } \frac{a + b}{2}$

Classifying Critical Points

Once we find $c$ where $f'(c) = 0$:

Example

For $f(x) = x^2 - 4x + 3$:

• $f'(x) = 2x - 4 = 0$ → $x = 2$
• $f''(2) = 2 > 0$ → local minimum
• $f(2) = 4 - 8 + 3 = -1$ is the minimum value

Global Extrema

On a closed interval $[a, b]$, the global max/min occurs either at a critical point or at an endpoint. Evaluate $f$ at all critical points and both endpoints, take the largest/smallest.

Your Task

Implement double critical_point(double (*f)(double), double a, double b, int n, double h) that uses bisection on $f'$ to find a critical point in $[a, b]$. Run $n$ bisection steps; $h$ is the step for numerical differentiation.