Critical Points and Where They Occur
Simple Explanation
A critical point of a function f is any x-value where f'(x) = 0 or where f'(x) does not exist. These are the only places a local maximum or local minimum can occur — the function's graph is momentarily flat (or has a sharp corner) there.
Why Do We Need It?
Critical points narrow down an infinite number of x-values to a short, finite list of candidates — turning "where is this function largest or smallest?" from an impossible search into a few calculations.
See It
A cubic curve that rises to a peak near x=−1, dips down to a valley near x=1, marked with dots at both points
Formula
The Definition of a Critical Point
A critical point of f occurs where f'(x) = 0, or where f'(x) is undefined
Critical points are the candidate locations for local maxima, local minima, or other notable features of a function's graph.
- f'(x)
- — the derivative of f
When to use it: Whenever locating a function's local extrema (maxima/minima) is the goal — always start by finding the critical points.
Worked Example
Find the critical points of a cubic
Find all critical points of f(x) = x³ − 3x.
Why Does This Work?
At a smooth local maximum or minimum, the tangent line must be exactly horizontal — any nonzero slope means the function is still rising or falling, and could not yet be at a peak or valley — so f'(x)=0 is a necessary condition. Where f'(x) does not exist (a sharp corner), the function can still turn around there, so those points must be checked too.
Real-Life Example
The peak height of a thrown ball
A ball's height over time rises, momentarily stops rising at its peak, then falls.
That exact instant — where the height function's derivative (its velocity) equals zero — is a critical point, and it is exactly where the ball reaches its maximum height.
Practice
Find the critical point of f(x) = x² − 4x + 1 (give the x-value).
EasyFor f(x) = x³ − 3x, what is the y-coordinate of the critical point at x = 1?
MediumCommon mistake
Substituting the critical x-value back into f'(x) instead of f(x) when finding the point's y-coordinate — f'(x) is exactly 0 there by construction, which is not the point's height.
Quick Review
- Critical points: where f'(x)=0 or f'(x) is undefined.
- They are the only candidates for local maxima and minima.
- Always substitute back into f (not f') to get the point's y-coordinate.