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Proof: strict convexity

From the discussion before equation \eqref{Deriv} we know that \(x(\lambda) \in C^1(\mathbb R)\) and hence that \(\lambda(x) \in C^1(\mathbb R)\). From Lemma \ref{freeEnergyEq} it follows that \(f''\) exists and is given by \(f''(x) = \lambda'(x)\). Recalling that a function is (strictly) convex if its second derivative is (strictly) positive, then from \eqref{Deriv} we know \(x'(\lambda) > 0\) for all \(\lambda \in \mathbb R\) and since \(\lambda(x)\) is the inverse

\[

f''(x) = \lambda'(x) > 0, \qquad \forall x \in \mathbb R. \qedhere

\]