Lef  f : R$\to \left(0,1\right)$ be a continuous function. Then which of the following functions (S) has (have) the value zero at some point in the interval?  

 A)

  $\begin{array}{l} f\left(x\right)\text{\hspace{0.17em}\hspace{0.33em} is positive\hspace{0.33em} and }{\int }_0^{\frac{\pi }{2}}f\left(t\right)\sin tdt\text{ is positive}\\ \\ \text{Hence, }f\left(x\right)+{\int }_0^{\frac{\pi }{2}}f\left(t\right)\sin tdt\text{is positive}\end{array}$

B) 

$\begin{array}{l}f\left(x\right)=x^9-f\left(x\right)\\ f\left(0\right)=0-f\left(0\right)=\text{ negative}\\ f\left(1\right)=1-f\left(1\right)=\text{positive}\\ \therefore f\left(x\right)=0foratleastonex\in \left(0,1\right)\end{array}$

C)

$\begin{array}{l}f\left(x\right)=x-{\int }_0^{\frac{\pi }{2}-x}f\left(t\right)\cos tdt\\ f\left(0\right)=0-{\int }_0^{\frac{\pi }{2}}f\left(t\right)\mathrm{cost}dt\text{\hspace{0.17em}is negative}\\ f\left(1\right)=1-f_0^{\frac{\pi }{2}-1}f\left(t\right)\cos tdtis \text{positive}\\ \therefore f\left(x\right)=0 foratleastonex\in \left(0,1\right)\end{array}$

D)

$\begin{array}{l}f\left(0\right)=e^0-{\int }_0^{0}f\left(t\right)\sin tdt=1\\ f\left(1\right)=e-{\int }_0^{1}f\left(t\right)\sin tdt=\text{Positve value}\\ f^1\left(x\right)=e^x-f\left(x\right)\sin x>0forx\in \left(0,1\right)\end{array}$