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

Options

A. `-(1)/(sqrt(3))`
B. `(1)/(sqrt(3))`
C. 0
D. none of these

Answer

A

Explanation

We have,
`f'(1)="tan"(pi)/(6)=(1)/(sqrt(3)), f'(2)= "tan"(pi)/(3)=sqrt(3) " and " f'(3)="tan"(pi)/(4)=1. `
` int_(2)^(3)f'(x)f''(x)dx+int_(1)^(3)f''(x)dx `
`=int_(2)^(3)f'(x)d(f'(x))+int_(1)^(3)d(f'(x)) `
`=[({f'(x)}^(2))/(2)]_(1)^(3) +[f'(x)]_(1)^(3) `
`=(1)/(2)[{f'(3)}^(2)-{f'(2)}^(2)] + [f'(3)-f'(1)] `
` =(1)/(2)(1-3)+(1-(1)/(sqrt(3)))=-(1)/(sqrt(3)) `