Second derivative and concavity
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Second Derivative Test
Use concavity to classify stationary points quickly and reliably.
The test
Once \(f'(c)=0\), the second derivative can often tell us the nature of the stationary point.
Maximum\[f''(c)<0\]
Minimum\[f''(c)>0\]
Inconclusive\[f''(c)=0\]
The test works because \(f''(x)\) describes concavity.
Worked example
Classify the stationary point of \(f(x)=x^2-4x+5\).
- \(f'(x)=2x-4\).
- \(2x-4=0\Rightarrow x=2\).
- \(f(2)=1\), so the point is \((2,1)\).
- \(f''(x)=2>0\), so \((2,1)\) is a minimum.
Important warning
If \(f''(c)=0\), do not automatically say “point of inflection”. The test is inconclusive. Use a sign table or a concavity check.
Using the second derivative test
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Second derivative worked example
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