
\begin{enumerate}
\item $\begin{aligned}[t]
\dfrac{1}{2}\Big[f(a+b)-\big(f(a)+f(b)\big)\Big]
& = \dfrac{1}{2}\Big[ (a+b)^2 - \big( a^2+b^2 \big) \Big]\\
& = \dfrac{1}{2}[a^2+2ab+b^2-a^2-b^2]\\
\Aboxed{\dfrac{1}{2}\Big[f(a+b)-\big(f(a)+f(b)\big)\Big] & = ab}
\end{aligned}$

\item $\begin{aligned}[t]
f(a+b)-f(a-b) & = (a+b)^2-  (a-b)^2\\
& = (a^2+2ab+b^2)-(a^2-2ab+b^2)\\
& = a^2+2ab+b^2-a^2+2ab-b^2\\
\Aboxed{f(a+b)-f(a-b) & = 4ab}
\end{aligned}$

\item $\begin{aligned}[t]
f\left(\frac{a+b}{2}\right)-f\left(\frac{a-b}{2}\right)
& = \left(\frac{a+b}{2}\right)^2 - \left(\frac{a-b}{2}\right)^2\\
& = \frac{(a+b)^2-(a-b)^2}{4}\\
& = \frac{4ab}{4}\text{ (d'après la question précédente)}\\
\Aboxed{f\left(\frac{a+b}{2}\right)-f\left(\frac{a-b}{2}\right)& = ab}
\end{aligned}$
\end{enumerate}
