If `f:R in R` is continuous and differentiable function such that
`int_(-1)^(x) f(t)dt+f'''(3) int_(x)^(1) dt=int_(1)^(x) t^(3)dt-f'(1)int_(0)^(x) t^(2)dt+f'(2) int_(x)^(3) r dt`, then the value of f'(4), is

Options

A. `48-8f'(1)+f'(2)`
B. `48-8f'(1)-f''(2)`
C. `48+8f'(1)-f'(2)`
D. none of these

Answer

B

Explanation

We have,
`overset(x)underset(-1)int f(t)dt+f'''(3)overset(0)underset(x)int dt`
`=overset(x)underset(1)int t^(3)dt-f'(1)overset(x)underset(0)int t^(2)dt+f'(2)overset(3)underset(x)int t dt`
`rArr overset(x)underset(-1)int f(t)dt-xf'''(3)=((x^(4))/(4)-(1)/(4))-f'(1)(x^(3))/(3)+f''(2)((9)/(2)-(x^(2))/(2))`
Differentiating w.r. to x, we get
`f(x)-f'''(3)=x^(3)-x^(2)f'(1)-xf''(2)`
Differentiating w.r. to x. we get
`f'(x)=3x^(2)-2xf'(1)-f''(2)`
`rArr f'(4)=48-8f'(1)-f''(2)`