Let $d \in X$ and set $$f: X \to \mathbb{R}: x \mapsto \left(\frac{1}{2}\right) \parallel x-d \parallel^2.$$ Use (*) to show that $f$ is strictly convex. Now let $g$ be any convex function. Show that $f + g$ is still strictly convex.
(*):= Let $A$ be a symmetric matrix in $\mathbb{R}^{n \times n}$ and set $$f(x):= \left(\frac{1}{2}\right) \langle Ax,x \rangle.$$ Let $x$ and $y$ be in $\mathbb{R}^n$ and $\lambda \in [0,1]$. Then $$(1-\lambda) f(x) + \lambda f(y) = f((1-\lambda)x+\lambda y) + \left(\frac{1}{2}\right) \lambda (1- \lambda) \langle A(x-y), x-y \rangle $$
I am unsure how to start this proof. Any suggestions would be appreciated