Critical Points
The interesting places on a curve — the peaks, the valleys, the moments it pauses — all
happen where the slope stops being uphill or downhill. A critical point is
an input x = c in the domain of f where
f'(c) = 0 \qquad\text{or}\qquad f'(c)\ \text{does not exist.}
These are the only places a smooth graph can turn around, so they are the first thing to find
when you want to understand a function's shape. We already know that
the sign of f' tells us where f rises and falls —
critical points are the boundaries between those intervals.
Where the tangent is flat
Slide the point along f(x) = x^3 - 3x and watch its tangent line.
Almost everywhere the tangent tilts, but at x = -1 and
x = 1 it goes perfectly horizontal — slope zero.
Those two inputs are the critical points. The readout turns green the instant
f'(x) = 0.
How to find them
The recipe is short. Differentiate, set the derivative to zero, and solve — then also note
any point where f' blows up.
f(x) = x^3 - 3x \;\Rightarrow\; f'(x) = 3x^2 - 3 = 3(x-1)(x+1) = 0 \;\Rightarrow\; x = \pm 1
Not every critical point is a peak or a valley, though. For
g(x) = x^3 we get g'(x) = 3x^2 = 0 at
x = 0, yet the curve only flattens for an instant and keeps
climbing — a critical point that is neither a maximum nor a minimum. Finding critical points
is step one; deciding what each is comes next.
Two kinds of critical point
A critical point can occur because the slope is zero (a smooth peak, valley, or flat
plateau) or because the slope is undefined — a sharp corner like
the one in |x| at x = 0, where the
curve abruptly changes direction. Step through both below.
Watch it on Khan Academy