DIFFERENTIATION 13

Stationary Points and Turning Points

Stationary points occur where the derivative is zero. They often identify maxima, minima or other important graph features.

The main idea

\[\boxed{\frac{dy}{dx}=0}\]

A stationary point has a horizontal tangent.

Solve \(f'(x)=0\), then find the corresponding function values.

For \(f(x)=x^2\), \(f'(x)=2x\), so the stationary point is \((0,0)\).

If the derivative changes from negative to positive, the point is a local minimum. If it changes from positive to negative, it is a local maximum.

\[-\rightarrow+\Rightarrow\text{minimum},\qquad +\rightarrow-\Rightarrow\text{maximum}.\]

At \(f'(a)=0\), if \(f''(a)\gt0\), the point is a local minimum. If \(f''(a)\lt0\), it is a local maximum.

\[\boxed{f'(a)=0,\ f''(a)\gt0\Rightarrow\text{minimum}}\]\[\boxed{f'(a)=0,\ f''(a)\lt0\Rightarrow\text{maximum}}\]

For \(f(x)=x^2-4x+7\), \(f'(x)=2x-4\), so \(x=2\). Since \(f''(2)=2\gt0\), \((2,3)\) is a minimum.

For \(f(x)=x^3\), \(f'(0)=0\), but the function increases on both sides. Thus the graph is stationary without turning.

If \(f''(a)=0\), inspect the sign of \(f'\) on either side or use other information.

\[\boxed{f'(x)=0\longrightarrow\text{candidate maximum, minimum or stationary inflection}}\]