Criteria for extrema
Criteria for extrema via the first and second derivative.
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Study Criteria for extrema
For \(f: (a,b)\to\mathbb{R}\) differentiable, \(x_0\in (a,b)\) stationary point:
…using the first derivative
Theorem 1.
If \(f'(x)\geq0\) for \(x<x_0\) and \(f'(x)\leq0\) for \(x>x_0\), then \(f\) has a local maximum in \(x_0\).
If \(f'(x)\leq0\) for \(x<x_0\) and \(f'(x)\geq0\) for \(x>x_0\), then \(f\) has a local minimum in \(x_0\).
…using the second derivative
Theorem 2. For \(f\) two times differentiable:
If \(f''(x_0)<0\), then \(f\) has a local maximum in \(x_0\).
If \(f''(x_0)>0\), then \(f\) has a local minimum in \(x_0\).
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