question_answer

The equation of a curve is y = f (x). The tangents at (1, f (1)), (2, f (2)) and (3, f (3)) make angles \[\frac{\pi }{6},\]\[\frac{\pi }{3}\] and \[\frac{\pi }{4}\]respectively with the positive direction of the x-axis. Then the value of \[\int_{2}^{3}{f'\,\,(x)f''\,\,(x)\,\,dx+\int_{1}^{3}{f''\,\,(x)\,\,dx}}\] is equal to

A)  \[-\frac{1}{\sqrt{3}}\]                           

B)  \[\frac{1}{\sqrt{3}}\]

C)  1                                

D)  \[\frac{2}{\sqrt{3}}\]

E)  None of these