Let `f(x)=x^3-3(a-2)x^2+3ax+7` and `f(x)` is increasing in `(0,1]` and decreasing is `[1,5)`, then roots of the equation `(f(x)-14)/((x-1)^2)=0` is (A) `1` (B) `3` (C) `7` (D) `-2`

Answer

7

Explanation

As f(x) is a polynomial function, hence continuous and diff `AA x in R`
Also f(x) increases in (0, 1) and decreases in (1.5)
Hence f'(1) = 0
`f'(x) = 3x^(2) - 3(a-2) 2x + 3a, f'(1) = 0, a = 5`
Hence `f(x) = x^(3) - 9x^(2) + 15x - 7, (f(x)-14)/((x-1)^(2)) = ((x-1)^(2)(x-7))/((x-1)^(2))` has solution x = 7