f : R → (0, 1)

(A)        Let ψ (x) = f(x) + $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$ 

                   ψ (0) = f(0) + $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$> 0

                   ψ (1) = f(1) + $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$ > 0

            $\therefore$ no root

(B)        Let h(x) = x⁹ – f (x)

            h (0) = – f (0) < 0

            and  h (1) = 1 – f (1) > 0 

(C)        Let g (x) = x – $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}-\mathrm{x}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{cost}\mathrm{dt}$

            g (0) = 0 – $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{cost}\mathrm{dt}$ < 0; g (1) = 1 – $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{cost}\mathrm{dt}>0$

            $\therefore$                                  g (x) = 0  for atleast one x $\in$ (0, 1)

(D)        Let f (x) = ex – $\underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$

            φ (0) = 1 – $\underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$; f (1) = e – $\underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}\left(\mathrm{t}\right)\mathrm{sint}\mathrm{dt}$