Home
Class 12
MATHS
If the function f givenn by f(x) =x^3-3...

If the function f givenn by `f(x) =x^3-3(a-2)x^2+3ax+7`, for some `a in R` is increasing in (0,1] and decreasing in [1,5), then a root of the equation `(f(x)-14)/((x-1)^2)=0(x ne1)` is

A

`-7`

B

`6`

C

`7`

D

`5`

Text Solution

Verified by Experts

The correct Answer is:
C
Given that function,
`f(x) = x ^(3) - 3 (a - 2) x ^(2) + 3a x + 7,` for some ` a in R` is increasing in `(0, 1]` and decreasing in `[1, 5)`
`f'(1) =0 " " [because ` tangent at x = 1 will be parallel to X- axis]
`rArr (3x^(2) - 6(a -2 ) x + 3 a) _(x = 1) = 0`
`rArr " " 3 -6 (a -2) + 3 a = 0`
`rArr " " 3 - 6a + 12 + 3a =0`
`rArr " " 15 - 3 a =0`
`rArr " " a = 5`
So, `f(x) = x ^(3) - 9x ^(2) + 15x + 7`
`rArr f(x) - 14 = x ^(3) - 9x ^(2) + 15x - 7`
`rArr f(x) - 14 = (x -1) (x ^(2) - 8x + 7) = (x -1)(x - 1) (x - 7)`
`rArr " " (f (x) - 14) /( (x - 1)^(2)) = (x - 7) " " `... (i)
Now, `" " (f(x) - 14) /( (x - 1)^(2)) =0,(x ne 1)`
`rArr " " x - 7 =0" " `[from Eq. (i)]
`rArr x = 7`
Promotional Banner

Similar Questions

Explore conceptually related problems

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

Find the roots of the equation (1)/(x)-(1)/(x-2)=3, x ne 0,2

For what values of a the function f given by f(x) = x^(2) + ax + 1 is increasing on [1, 2]?

Find the intervals in which the function f given by f(x) =x^(3) +(1)/(x^(3)) , xne 0 is (i) increasing (ii) decreasing .

Find the intervals in which the function f given by f(x) = 2x^(3) – 3x^(2) – 36x + 7 is (a) increasing (b) decreasing

Let a function f be defined by f(x)=(x-|x|)/x for x ne 0 and f(0)=2. Then f is

Show that the function f given by f(x)= {{:(x^3+3," if "x ne 0),(1," if "x=0):} is not continuous at x=0.

Find the range of the function f(x)=cot^(-1)(log)_(0. 5)(x^4-2x^2+3)

Let f be the function f(x)=cosx-(1-(x^2)/2)dot Then f(x) is an increasing function in (0,oo) f(x) is a decreasing function in (-oo,oo) f(x) is an increasing function in (-oo,oo) f(x) is a decreasing function in (-oo,0)

If f(x)=xe^(x(x−1)) , then f(x) is (a) increasing on [−1/2,1] (b) decreasing on R (c) increasing on R (d) decreasing on [−1/2,1]