Home
Class 12
MATHS
Let f(x) be the fourth degree polynomial...

Let `f(x)` be the fourth degree polynomial such that `f^(prime)(0)-6,f(0)=2a n d(lim)_(xvec1)(f(x))/((x-1)^2)=1` The value of `f(2)` is `3` b. 1 c. `0` d.`2`

A

4

B

5

C

6

D

7

Text Solution

Verified by Experts

The correct Answer is:
C
Since `underset(xrarr1)(lim)(f(x))/((x-1)^(2))=1,f(1)=0`
`therefore" "underset(xrarr1)(lim)(f(x))/((x-1)^(2))=underset(xrarr1)(lim)(f'(x))/(2(x-1))=1`
`rArr" "f'(1)=0`
`therefore" "underset(xrarr1)(lim)(f''(x))/(2)=1rArrf''(1)=2`
Since x = 1 is root of f(x) = 0 and `f'(x) = 0.`
`f(x)=(x-1)^(2)(ax^(2)+bx+2)" "(because f(0)=2)`
`rArr" "f'(x)=2(x-1)(ax^(2)+bx+2)+(2ax+b)(x-1)^(2)`
`because" "f'(0)=-6rArr b=-2`
Using f''(1)= 2, we get `a+b=-1 rArr a=1`
`rArr" "f(x)=(x-1)^(2)(x^(2)-2x+2)`
`rArr" "f(x)=(x-1)^(4)+(x-1)^(2)`
`rArr" "f(2)=1+1=2`
`"Also, "f'(x)=4(x-1)^(3)+2(x-1)`
`rArr" "f'(2)=4+2=6`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|2 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise Comprehension Type|4 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|24 Videos

  • JEE 2019

    CENGAGE PUBLICATION|Exercise Chapter 10|8 Videos

  • LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS

    CENGAGE PUBLICATION|Exercise DPP 1.2|10 Videos

Similar Questions

Explore conceptually related problems

Let ("lim")_(xvec1)(x^a-a x+a-1)/((x-1)^2)=f(a)dot Then the value of f(4) is _________

Let f(x) be a polynomial of degree 3 such that f(3)=1,f^(prime)(3)=-1,f^('')(3)=0,a n df^(''')(3)=12. Then the value of f^(prime)(1) is (a) 12 (b) 23 (c) -13 (d) none of these

Let f(x) be a polynomial satisfying lim_(xtooo) (x^(2)f(x))/(2x^(5)+3)=6" and "f(1)=3,f(3)=7" and "f(5)=11. Then The value of f(0) is

A polynomial f(x0 satisfies the condition f(x).f(1/x)=f(x)+f(1/x) and f(10) = 1001, then the value of f(2) =?

Find the degree two polynomial function f(x) for which it is known that f(0) = 1, f(1) = 5, f(2) = 11.

Suppose |[f'(x),f(x)],[f''(x),f'(x)]|=0 is continuously differentiable function with f^(prime)(x)!=0 and satisfies f(0)=1 and f'(0)=2 then lim_(x->0)(f(x)-1)/x is a. 1//2 b. 1 c. 2 d. 0

Let f(x) be a function defined on (-a ,a) with a > 0. Assume that f(x) is continuous at x=0a n d(lim)_(xvec0)(f(x)-f(k x))/x=alpha,w h e r ek in (0,1) then a. f^(prime)(0^+)=0 b. f^(prime)(0^-)=alpha/(1-k) c. f(x) is differentiable at x=0 d. f(x) is non-differentiable at x=0

Let f(x) be continuous functions f: RvecR satisfying f(0)=1a n df(2x)-f(x)=xdot Then the value of f(3) is 2 b. 3 c. 4 d. 5

Let f (x+(1)/(x))= x^(2) +(1)/(x^(2)) , x ne 0, then the value of f (x) is-

Let f(x y)=f(x)f(y)AAx , y in R and f is differentiable at x=1 such that f^(prime)(1)=1. Also, f(1)!=0,f(2)=3. Then find f^(prime)(2)