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