Coefficient combination from a local minimum

Let $f:\left[0,\frac\pi2\right]\to\mathbb R$ be defined as $f(x)=\alpha x+\beta\sin x$, where $\alpha,\beta\in\mathbb R$. Let $f$ have a local minimum at $x=\frac\pi4$ with $f\left(\frac\pi4\right)=\frac{\pi-4}{4\sqrt2}$. Then $8\sqrt2\,\alpha+4\beta$ equals __________________