Locate stationary points and justify their nature.
Notes and key methods
A differentiable curve has a stationary point where \(\displaystyle f'(x)=0\). Solve for the abscissae, then find the ordinates using f, not f′. If f″(a)>0 there is a local minimum; if f″(a)<0 there is a local maximum.
If f″\(\displaystyle (a)=0\), this test is inconclusive: inspect the sign of f′ on both sides or use further reasoning. A stationary inflection is neither a maximum nor a minimum.
Check the conditions
Give coordinates, not just stationary abscissae.
A zero second derivative needs another test.
Three worked examples
Example 1 · Stationary points and second derivatives
Consider \(y=- 12 x + \left(x - 5\right)^{3} + 65\).
Use the labelled scales alongside the numerical method.
Find both stationary points.
Classify both stationary points with justification.
Find the height of the local maximum.
Find the height of the local minimum.
Show the reasoning
(a)
\(3 \left(x - 5\right)^{2} - 12\) vanishes at \(3\) and \(7\). Substitute each into f.
Answer: \(\left( 3, \ 21\right)\) and \(\left( 7, \ -11\right)\)
(b)
The second derivative is \(6 \left(x - 5\right)\). Its values are \(-12\) and \(12\), respectively.
Answer: The first is a local maximum; the second is a local minimum.
(c)
Evaluate f at h−d; the second derivative is negative.
\(21\)
Answer: \(21\).
(d)
Evaluate f at h+d; the second derivative is positive.
\(-11\)
Answer: \(-11\).
Example 2 · Stationary points and second derivatives
