Critical Points, Maxima and Minima
Understanding how calculus reveals the peaks, valleys, and turning points of functions is one of the most powerful applications of derivatives.
1. What is a Critical Point?
- $f'(c) = 0$ (the derivative equals zero — the tangent line is horizontal), or
- $f'(c)$ does not exist (the derivative is undefined at that point).
Critical points are the candidates where a function might change from increasing to decreasing (or vice versa). Think of them as the places where the function "pauses" before potentially changing direction.
Why do we care?
Every local maximum and local minimum of a differentiable function occurs at a critical point. However, not every critical point is a max or min — some are inflection points (like $x = 0$ for $f(x) = x^3$).
2. Local (Relative) vs. Global (Absolute) Extrema
Local Maximum
$f(c)$ is a local maximum if $f(c) \geq f(x)$ for all $x$ near $c$. It's the highest point in its immediate neighborhood.
Local Minimum
$f(c)$ is a local minimum if $f(c) \leq f(x)$ for all $x$ near $c$. It's the lowest point in its immediate neighborhood.