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What is Calculus I

a Critical Point?

A point where the derivative is zero or undefined — the candidates for maxima, minima, and saddle points of a function.

Simple Definition

A critical point of a function \(f\) is any value \(x = c\) in its domain where either \(f'(c) = 0\) or \(f'(c)\) does not exist. These are the only places where a local maximum or minimum can occur — so finding critical points is the first step in every optimisation problem.

Think of it this way: at a peak or a valley, the tangent line is perfectly horizontal — slope zero. At a sharp corner (like the tip of \(|x|\)), the derivative doesn't exist. Both are critical points. Neither one is guaranteed to be a max or min — that requires further testing.

Three Outcomes

⛰️

Local Maximum

Function is higher here than at all nearby points. Derivative goes + → 0 → −.

🏔️

Local Minimum

Function is lower here than at all nearby points. Derivative goes − → 0 → +.

〰️

Saddle Point

Derivative is zero but it's neither a max nor a min. Sign of f' doesn't change.

Intuition

🏔️

Standing at the top of a hill

At the peak of a hill, the ground is momentarily flat — zero slope in every direction. That's a critical point. Walking down the other side, the slope is negative. So the derivative went from positive (climbing) to zero (peak) to negative (descending). The sign change confirms it's a maximum.

⚠️

Critical ≠ extreme

Not every critical point is a max or min. Consider \(f(x) = x^3\) at \(x = 0\): \(f'(0) = 0\), so it's a critical point — but the function keeps increasing through zero. It's a saddle point (also called an inflection point with zero slope). Always test after finding critical points.

Visual — Critical Points Labelled

Green = local max · Red = local min · Gold = saddle point · Tangent lines shown at each critical point

How to Find Critical Points — Step by Step

Find \(f'(x)\)

Differentiate the function using whatever rules apply — power rule, chain rule, product rule, etc.

Set \(f'(x) = 0\) and solve

These give the "stationary points" — where the tangent line is horizontal.

Find where \(f'(x)\) is undefined

Check for division by zero, square roots of negatives, or other domain issues in \(f'\). These are also critical points if \(x\) is in the domain of \(f\).

Classify using first or second derivative test

Determine whether each critical point is a local max, local min, or saddle point.

The Two Classification Tests

First Derivative Test

Check the sign of f' around c

+ → −: local maximum
− → +: local minimum
no sign change: saddle point

Works even when the second derivative doesn't exist.

Second Derivative Test

Evaluate f″(c)

f″(c) < 0: local maximum (concave down)
f″(c) > 0: local minimum (concave up)
f″(c) = 0: inconclusive — use first derivative test

Faster when f″ is easy to compute.

Quick Examples

Example 1

\(f(x) = x^3 - 3x\)

\(f'(x) = 3x^2 - 3 = 0\)
\(x = \pm 1\)
\(f''(1) = 6 > 0\) → min at \(x=1\)
\(f''(-1) = -6 < 0\) → max at \(x=-1\)

Example 2 — Saddle

\(f(x) = x^3\)

\(f'(0) = 0\) → critical point
\(f''(0) = 0\) → inconclusive
First deriv test: + → + (no change)
→ saddle point at \(x = 0\)

Example 3 — Undefined f'

\(f(x) = |x|\)

\(f'(0)\) does not exist (sharp corner)
→ critical point at \(x = 0\)
First deriv: − → + → local min

Example 4 — Full process

\(f(x) = x^4 - 4x^2\)

\(f'= 4x^3 - 8x = 4x(x^2-2) = 0\)
\(x = 0, \pm\sqrt{2}\)
Test each: max at 0, minima at \(\pm\sqrt{2}\)

Common Mistakes

✗ Assuming every critical point is a max or min

Critical points are only candidates. A saddle point satisfies \(f'(c) = 0\) but is neither a max nor a min. Always classify after finding the critical points — never assume.

✗ Forgetting critical points where f' is undefined

Setting \(f'(x) = 0\) finds stationary points, but there may also be critical points where \(f'\) doesn't exist. Check the denominator of \(f'\), any roots, and other potential undefined spots.

✗ Confusing local and global extrema

A local max is just the highest point in a neighbourhood — not necessarily the highest point overall. To find the global max on \([a,b]\), compare all local maxima and also evaluate \(f\) at the endpoints \(a\) and \(b\).

Related Concepts

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