Fermat’s theorem on stationary points

What you are looking at

The yellow line is the tangent at $P$, and the working gives $f'(x_0)$ and whether $P$ is a local maximum, a local minimum or neither. On $f(x) = x^3 - 3x$ the tangent flattens at $x = -1$ and $x = 1$, exactly where the maximum and the minimum are: $f'(x) = 3x^2 - 3 = 0$ there.

The theorem only works one way. On $f(x) = x^3$ the tangent is horizontal at $0$, but the graph keeps rising: $f'(0) = 0$ without an extremum. On $f(x) = |x^2 - 1|$ there are minima at $x = \pm 1$ where the derivative does not exist — the one-sided derivatives are shown instead. Critical points are both kinds: where $f' = 0$, and where $f'$ does not exist.

Try this

1. 1On $x^3 - 3x$, drag $P$ to $x = -1$ and then to $x = 1$: $f'(x_0) = 0$ and the working names the extremum.
2. 2Switch to $x^3$ and drag $P$ to $0$: a horizontal tangent, and no extremum.
3. 3Switch to $|x^2 - 1|$: at $x = 1$ there is a minimum but no tangent — the left and right derivatives are $-2$ and $2$.
4. 4Press Sweep and watch the tangent turn as $P$ travels: it is horizontal only at the critical points.

Mistakes students make

• Using the converse: $f'(x_0) = 0$ does not mean an extremum ($x^3$ at $0$). You still need a change of sign of $f'$, or the second derivative.
• Looking for extrema only where $f' = 0$: corners are candidates too ($|x|$ at $0$), and so are the endpoints of a closed interval.
• Applying it at an endpoint: at $\alpha$ or $\beta$ a maximum can have $f' \ne 0$ — the point must be interior.

In the exam

• The proof is short and worth knowing: at an interior local maximum, $\dfrac{f(x) - f(x_0)}{x - x_0}$ is $\ge 0$ to the left of $x_0$ and $\le 0$ to the right, so $f'(x_0)$, which exists, is both $\ge 0$ and $\le 0$.
• AP defines critical points as the points where $f' = 0$ or $f'$ does not exist — exactly the set this theorem points to.

Where it is taught

• AP Calculus AB/BC · Unit 5 (5.2 critical points)
• IB Maths AA SL/HL · 5.8 (local maxima and minima)
• Edexcel A-Level Pure Year 1 · Differentiation (stationary points)
• Greek Γ΄ Λυκείου · §2.7