Home
Class 12
MATHS
If a differentiable function satisfies (...

If a differentiable function satisfies `(x-y) f (x+y) -(x+y)f (x-y) =2 (x ^(2)y- y ^(2)) AA x, y in R and f (1)=2,` then:

A

f(x) must be polynomial function

B

f(3) = 12

C

f(0) = 0

D

f(x) may not be differentiable

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(x-y)f(x+y)-(x+y)f(x-y)=2y(x-y)(x+y)`
Let `x-y=u, x+y=v`. Then
`uf(v)-vf(u)=uv(v-u)`
`rArr" "(f(v))/(v)-(f(u))/(u)=v-u`
`rArr" "((f(v))/(x)-v)=((f(u))/(u)-u)=" constant"`
`"Let "(f(x))/(x)-x=lambda. " Then"`
`f(x)=(lambdax+x^(2))`
Since `f(1)=2" then "lambda=1`
`therefore" "f(x)=x^(2)+x`
Promotional Banner

Topper's Solved these Questions

  • FUNCTIONS

    CENGAGE ENGLISH|Exercise Comprehension Type|7 Videos

  • FUNCTIONS

    CENGAGE ENGLISH|Exercise Comprehension Type|7 Videos

  • EQUATION OF PLANE AND ITS APPLICATIONS -II

    CENGAGE ENGLISH|Exercise DPP 3.4|14 Videos

  • GETTING STARTED WITH GRAPHS

    CENGAGE ENGLISH|Exercise Exercises 1.18|1 Videos

Similar Questions

Explore conceptually related problems

If a function satisfies (x-y)f(x+y)-(x+y)f(x-y)=2(x^2 y-y^3) AA x, y in R and f(1)=2 , then

If a function satisfies (x-y)f(x+y)-(x+y)f(x-y)=2(x^2 y-y^3) AA x, y in R and f(1)=2, then a) f(x) must be polynomial function, b) f(3)=12, c) f(0)=0, d) f(x) may not be differentiable.

If f is a real- valued differentiable function satisfying |f(x) - f(y)| le (x-y)^(2) ,x ,y , in R and f(0) =0 then f(1) equals

Let f(x+y)+f(x-y)=2f(x)f(y) AA x,y in R and f(0)=k , then

If f(x+y) = f(x) + f(y) + |x|y+xy^(2),AA x, y in R and f'(0) = 0 , then

Let 'f' be a fifferentiable real valued function satisfying f (x+2y) =f (x) +f (2y) + 6xy (x+2y) AA x, y in R. Then f ' (0), f" (1), f'(2)….. are in

Let f : R rarr R be a differentiable function satisfying f(x) = f(y) f(x - y), AA x, y in R and f'(0) = int_(0)^(4) {2x}dx , where {.} denotes the fractional part function and f'(-3) = alpha e^(beta) . Then, |alpha + beta| is equal to.......

Let f be a differentiable function satisfying the condition f ((x)/(y)) = (f(x))/(f (y)) (y ne 0, f (y) ne 0) AA x, y in R and f '(1) =2. If the smaller area enclosed by y = f(x) , x ^(2)+y^(2) =2 is A, then findal [A], where [.] represents the greatest integer function.

f (x) is differentiable function satisfying the relationship f ^(2) (x) + f ^(2) (y)+2 (xy-1) = f^(2) (x+y) AA x, y in R Also f (x) gt 0 AA x in R and f (sqrt2)=2. Then which of the following statement (s) is/are correct about f (x) ?

If f' ((x)/(y)). f((y)/(x))=(x^(2)+y^(2))/(xy) AA x,y in R^(+) and f(1)=1 , then f^(2) (x) is