How to prove these integral inequalities? a) $f(x)>0$ and $f(x)\in C[a,b]$
Prove $$\left(\int_a^bf(x)\sin x\,dx\right)^2 +\left(\int_a^bf(x)\cos x\,dx\right)^2 \le \left(\int_a^bf(x)\,dx\right)^2$$
I have tried Cauchy-Schwarz inequality but failed to prove.
b) $f(x)$ is differentiable in $[0,1]$
Prove 
$$|f(0)|\le \int_0^1|f(x)|\,dx+\int_0^1|f'(x)|dx$$
Any Helps or Tips,Thanks

Hint: For part a), use Jensen's inequality with weighted measure $f(x)\,\mathrm{d}x$. Since $f(x)>0$, Jensen says that for a convex function $\phi$
$$
\phi\left(\frac1{\int_Xf(x)\mathrm{d}x}\int_Xg(x)\,f(x)\mathrm{d}x\right)
\le\frac1{\int_Xf(x)\mathrm{d}x}\int_X\phi(g(x))\,f(x)\mathrm{d}x
$$
Hint: For part b), note that for $x\in[0,1]$,
$$
f(0)-f(x)\le\int_0^1|f'(t)|\,\mathrm{d}t
$$
and integrate over $[0,1]$.
