Stationary points and their nature

A typical question: Find the coordinates of the stationary points of $y = 2x^3 - 3x^2 - 12x + 7$, and determine their nature.

Giveaway words: stationary point, turning point, determine the nature, maximum point, minimum point.

A stationary point is where the curve is flat for a moment, so the gradient is zero: $\dfrac{dy}{dx} = 0$.

• A maximum is the top of a hill.
• A minimum is the bottom of a valley.
• "Nature" means: which one is it?
• (A third kind, a stationary point of inflexion like $y = x^3$ at $x = 0$, is beyond this syllabus.)

$\dfrac{d^2y}{dx^2}$ is the second derivative: differentiate $\dfrac{dy}{dx}$ again. It tells you how the gradient is changing.

The same five steps every time:

1. Differentiate: find $\dfrac{dy}{dx}$.
2. Solve: set $\dfrac{dy}{dx} = 0$ and find the $x$-values.
3. Find y: put each $x$ into the original equation, not the derivative.
4. Differentiate again: find $\dfrac{d^2y}{dx^2}$ and put each $x$ in.
5. Decide: negative means maximum, positive means minimum.

Memory hook for step 5: negative is a frown ∩ (a hilltop). Positive is a smile ∪ (a valley).

Why? A positive second derivative means the gradient is increasing, from $-$ to $+$: the curve went down, then up. A negative one means the gradient goes from $+$ to $-$: up, then down.