Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$
where:
1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and
2). $0\leqslant f(x)\in C^{\infty}$ is monotonically decreasing in $[0,T]$;
Suppose:
$\lim\limits_{x\to 0}F(x)=0,\quad\text{and } \exists\; x_0\in(0,\dfrac{\pi}{2T})\text{ such that}:F(x_0)=0.$
Prove:
$x_0$ is the unique root of $F(x)=0 $ in $(0,\dfrac{\pi}{2T})$;
or equivalently:
$F(x)>0,\;\forall\, x_0<x<\dfrac{\pi}{2T},$ and $F(x)<0,\;\forall\; 0<x<x_0$
or is there any counter example?