12.9PureStretch

Stationary Points

A stationary point is where a curve is momentarily flat — gradient zero. Finding them and classifying them (maximum, minimum or point of inflection) is one of the most examined skills in the whole course.

30 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.9 Stationary Points (Maximum and Minimum)Watch the full walkthrough before the notes below.

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What you'll be able to do

Find stationary points by solving dy/dx = 0
Classify them using the second derivative
Classify them using a gradient sign table
Find the coordinates of each stationary point

Finding stationary points

At a stationary point the gradient is zero. So set $\frac{d y}{d x} = 0$ and solve for $x$ , then substitute back into $y$ for the coordinates.

\frac{d y}{d x} = 0 \Rightarrow stationary point

Solve for

x

, then find

y

\frac{d y}{d x} = 2 x - 4 = 0 \Rightarrow x = 2

2At

x = 2

y = 4 - 8 + 1 = - 3

Answer

(2, - 3)

Classifying with the second derivative

Substitute the stationary $x$ into $\frac{d ^{2} y}{d x ^{2}}$ . Positive means a minimum ( $\cup$ ); negative means a maximum ( $\cap$ ). If it is zero, the test is inconclusive — use a sign table.

f^{''} (x) > 0 \Rightarrow minimum, f^{''} (x) < 0 \Rightarrow maximum

The second-derivative test.

\frac{d ^{2} y}{d x ^{2}} = 2 > 0

Answerminimum

Tip — Positive second derivative = minimum (∪-shaped); negative = maximum (∩-shaped).

The gradient sign table

Alternatively, check the sign of $\frac{d y}{d x}$ just before and just after the point. Going $+ \to 0 \to -$ is a maximum; $- \to 0 \to +$ is a minimum; the same sign on both sides is a point of inflection.

Formula recap

\frac{d y}{d x} = 0

Locate stationary points.

f^{''} (x) > 0 \Rightarrow min, f^{''} (x) < 0 \Rightarrow max

Second-derivative test.

+ \to 0 \to - : max, - \to 0 \to + : min

Sign-table test.

Common mistakes to avoid

Stopping at the x-value and not finding y.

A stationary point is a coordinate — substitute back for y.

Assuming a positive second derivative means a maximum.

Positive ⟶ minimum; negative ⟶ maximum.

Key takeaways

Stationary points: solve dy/dx = 0, then find y.
Second-derivative test: f″ > 0 minimum, f″ < 0 maximum.
Or use a gradient sign table around the point.

Test yourself

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