Calculus

Stationary Points

Identify and classify local maxima, local minima and horizontal points of inflection.

Applications overviewSecond derivative testPoints of inflection

Key idea

A stationary point occurs where the tangent to the curve is horizontal, so the derivative is zero.

\[ f'(x)=0 \]

This condition finds candidates. The point might be a local maximum, a local minimum, or a horizontal point of inflection.

local maxlocal min
At a local maximum the gradient changes from positive to negative. At a local minimum it changes from negative to positive.

Worked example

Find and classify the stationary points of \(f(x)=x^3-3x^2-9x+4\).

  1. Differentiate: \(f'(x)=3x^2-6x-9\).
  2. Solve \(f'(x)=0\): \(3x^2-6x-9=0\Rightarrow x^2-2x-3=0\Rightarrow x=-1,3\).
  3. Find the coordinates: \(f(-1)=9\), \(f(3)=-23\).
  4. Use the second derivative: \(f''(x)=6x-6\).

At \(x=-1\), \(f''(-1)=-12<0\), so \((-1,9)\) is a local maximum. At \(x=3\), \(f''(3)=12>0\), so \((3,-23)\) is a local minimum.

Related videos

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Stationary points: introduction

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Classifying stationary points

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Stationary points worked example

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