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

I think for case a) double integral should help you.
$$
\left(\int_a^bf(x)\sin x\,dx\right)^2 +\left(\int_a^bf(x)\cos x\,dx\right)^2 = \int_a^b\int_a^bf(x)f(y)sinxsinydxdy+\int_a^b\int_a^bf(x)f(y)cosxcosydxdy =\int_a^b\int_a^bf(x)f(y)(sinxsiny+cosxcosy)dxdy=\int_a^b\int_a^bf(x)f(y)cos(x+y)dxdy\le \int_a^b\int_a^bf(x)f(y)dxdy=\left(\int_a^bf(x)\,dx\right)^2
$$
