Home
Class 12
MATHS
f(x) is a polynomial function, f: R rarr...

f(x) is a polynomial function, `f: R rarr R,` such that `f(2x)=f'(x)f''(x).`
`f(3)` is

A

4

B

12

C

15

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Suppose degree of f(x) = n. Then degree of f'=n-1 and degree of `f''=n-2.` So,
`n=n-1+n-2`
Hence, n=3.
So, put `f(x)=ax^(3)+bx^(2)+cx+d (where a ne 0).`
From `f(2x)=f'(x).f''(x),` we have
`8ax^(3)+4bx^(2)+2cx+d=(3ax^(2)+2bx+c)(6ax+2b)`
`=18a^(2)x^(3)+18abx^(2)+(6ac+4b^(2))x+2bc`
Comparing coefficients of terms, we have
`18a^(2)=8a" "rArra=4//9`
`18ab=4b" "rArrb=0`
`2c=6ac+4b^(2)" "rArrc=0`
`d=2bc" "rArrd=0`
`"or "f(x)=(4x^(3))/(9),` which is clearly one-one and onto
`"or "f(3)=12`
`"Also, "(4x^(3))/(9)=xor x=0, x= pm 3//2.`
Hence, sum of roots of equation is zero.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Matrix Match Type|6 Videos

  • DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Numerical Value Type|45 Videos

  • DIFFERENTIATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|29 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE PUBLICATION|Exercise All Questions|578 Videos

  • DOT PRODUCT

    CENGAGE PUBLICATION|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

f(x) is a polynomial function, f: R rarr R, such that f(2x)=f'(x)f''(x). f(x) is (A) one-one and onto (B) one-one and into (C) many-one and onto (D) many-one and into

f(x) is a polynomial function, f: R rarr R, such that f(2x)=f'(x)f''(x). Equation f(x) = x has (A) three real and positive roots (B) three real and negative roots (C) one real root (D) three real roots such that sum of roots is zero

Knowledge Check

  • Let f be a polynomial function such that f(x) f(y) + 2 = f(x) + f(y) + f(xy) AA x,y in R^+ cup {0} and f(x) is one-one AA x,y in R^+ with f(0)=1 and f(1)=2 Answer the following questions based on above passage: The function y=f(x) is given by

    A
    `x^(1//3)-1`
    B
    `1+(2x^3)/3`
    C
    `1+x^2`
    D
    `1-x^2`

  • If f: RR^(+) rarr RR ^(+) is a polynomial function satisfying the functional equation f{f(x)}=6x-f(x) , then f(17) is equal to ___

    A
    17
    B
    -15
    C
    34
    D
    -34

    • Similar Questions

    Explore conceptually related problems

    Let A={x in R:x" is not a positive integer "} define a function f:AtoR" such that "f(x)=(2x)/(x-1) . Then f is

    Let f: \mathbb{R} rarr \mathbb{R} be a twice differentiable function such that f(x+pi)=f(x) and f''(x) + f(x) geq 0 for all x in \mathbb{R} . Show that f(x) geq 0 for all x in \mathbb{R} .

    Determine all functions f : R rarr R satisfying f(x-f(y)) = f(f(y)) + xf(y) + f(x) - 1 AA x, y in R

    If f(x) is a polynomial satisfying f(x)f(1//x)=f(x) +f(1//x) and f(3)=28, then f(4) is equal to

    If f: R^+rarrR^+ is a polynomial function satisfying the functional equation f(f(x))=6x-f(x),t h e nf(17) is equal to 17 (b) 51 (c) 34 (d) -34

    If a function f: R ->R be such that f(x-f(y)) = f(f(y) )+xf(y)+f(x) -1 AA x , y in R then f(2)=

    Home
    Profile