Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. Justify that $f$ is of class $\mathcal{C}^2$, decreasing and convex on $I$.
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Justify that $f$ is of class $\mathcal{C}^2$, decreasing and convex on $I$.