JEE Advanced 2017 Power

Let $f: \mathbb {R \to (0,1)}$ be a continuos function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$ ?

A. $x^{9}-f(x)$

B. $\displaystyle f(x)+\int_{0}^{\frac{\pi}{2}}f(t)\sin(t) \ dt$

C. $\displaystyle x-\int_{0}^{\frac{\pi}{2}-x}f(t)\cos(t) \ dt$

D. $\displaystyle e^{x}-\int_{0}^{x}f(t)\sin(t) \ dt$