Characterization of convexity/concavity via the second derivative.
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Second derivative and
convexity/concavity
Definition 1. For \(D\) interval, \(f: D\to\mathbb{R}\):
\(f\) convex, if
for all \(x_1,x_2\in D\) and \(\lambda \in (0,1)\): \[f(\lambda x_1+(1-\lambda)x_2)\leq\lambda
f(x_1)+(1-\lambda)f(x_2).\]
\(f\) strictly
convex if “\(<\)” instead of
“\(\leq\)”.
\(f\) (strictly)
concave, if \(-f\) (strictly)
convex, i.e. “\(\geq\)” (“\(>\)”) instead of “\(\leq\)” (“\(<\)”).
For \(f\) two times
differentiable:
\(f''(x)\geq 0\) for all
\(x\in D\) \(\iff\) \(f\) convex.
\(f''(x)>0\) for all
\(x\in D\) \(\ \implies\) \(f\) strictly convex.
\(f''(x)\leq 0\) for all
\(x\in D\) \(\iff\) \(f\) concave.
\(f''(x)<0\) for all
\(x\in D\) \(\ \implies\) \(f\) strictly concave.
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