A differentiable function satisfies `f(x) = int_(0)^(x) (f(t) cot t - cos(t - x))dt`.
Which of the following hold(s) good?

Options

A. f(x) has a minimum value 1 - e
B. f(x) has a maximum value 1 - `e^(-1)`
C. `f''((pi)/(2))=e`
D. `f'(0) = 1`

Answer

A, B, C

Explanation

`f(x) = underset(0)overset(x)int {f(t) cos t - cos(t-x)}dx`
`= underset(0)overset(x)int f(t) cos t dt -underset(0)overset(x) int cos(-t)dt" "["using"underset(0)overset(a)intf(x)dx=underset(0)overset(a)int f(a-x)dx]`
`therefore" "f(x) = underset(0)overset(x) int f(t) cos t dt - sin x`
Differentiate both sides w.r.t.x, f'(x) = f(x) cos x - cos x
`or" "(dy)/(dx)-y cos x = -cos x`
`therefore" "I.F. = e^(-int cos x dx) = e^(-sin x)`
Hence, solution is `y e^(-sin x) = - int e^(-sin x) cos x dx`
`or" "y e^(-sin x) = C + e^(-sin x)`
`or" "y = C e^(sin x) + 1`
At `x = 0, y = 0 rArr C = -1`
`therefore" "f(x) = 1 - e^(sin x)`
`" "f_(min)=1-e" "("when x" = pi//2)`
and`" "f_(max) = 1 -e^(-1)" "("when x" = -pi//2)`
`" "f'(x) = -e^(sin x) cos x`
`rArr" "f'(0) = -1`
Also `f"(x) = - [cos^(2) x e^(sin x)-e^(sin x) * sin x]`
`f"((pi)/(2))=e`