Home
Class 12
MATHS
If a function satisfies (x-y)f(x+y)-(x+y...

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

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|Exercise Comprehension Type|7 Videos

  • FUNCTIONS

    CENGAGE|Exercise Single Correct Answer Type|56 Videos

  • EQUATION OF PLANE AND ITS APPLICATIONS -II

    CENGAGE|Exercise DPP 3.4|19 Videos

  • GETTING STARTED WITH GRAPHS

    CENGAGE|Exercise Exercise|18 Videos

Similar Questions

Explore conceptually related problems

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

If f is polynomial function satisfying 2+f(x)f(y)=f(x)+f(y)+f(xy)AA x,y in R and if f(2)=5, then find the value of f(f(2))

Let f(x) be a differentiable function satisfying f(y)f((x)/(y))=f(x)AA,x,y in R,y!=0 and f(1)!=0,f'(1)=3 then

If the function / satisfies the relation f(x+y)+f(x-y)=2f(x),f(y)AA x,y in R and f(0)!=0 ,then f(x) is an even function f(x) is an odd function If f(2)=a, then f(-2)=a If f(4)=b, then f(-4)=-b

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

Let f:R rarr R be a differentiable function satisfying 2f((x+y)/(2))-f(y)=f(x) AA x , y in R ,if f(0)=5 and f'(0)=-1 , then f(1)+f(2)+f(3) equals

A real valued function f(x) is satisfying f(x+y)=yf(x)+xf(y)+xy-2 for all x,y in R and f(1)=3, then f(11) is

If f(x) is a function satisfying f(x)-f(y)=(y^(y))/(x^(x))f((x^(x))/(y^(y)))AA x,y in R^(*) and f'(1)=1, then find f(x)

Let f be a real valued differentiable function satisfying f(x+y)=f(x)+f(y)-xy-1,AA x,y in R and f(1)=1. Equation ofthe normal to the graph of f(x) at the point where y=f(x) cuts the y-axis,is

CENGAGE-FUNCTIONS-Question Bank
  1. If a function satisfies (x-y)f(x+y)-(x+y)f(x-y)=2(x^2 y-y^3) AA x, y i...

    Text Solution

    |

  2. If f(x)=2 x-1 then number of solution (s) of the equation |f(x)|=|f(|x...

    Text Solution

    |

  3. If there are four distinct real numbers satisfying thẻ equation |x^2-4...

    Text Solution

    |

  4. Let f: R rarr R be a polynomial function satisfying f(f(x) -2 y)=2 x-3...

    Text Solution

    |

  5. The least integral value in the range of f(x)=e^(2 x+1/(1+e^2 x) i...

    Text Solution

    |

  6. Let P(x)=x^4+a x^3+b x^2+c x+d be a polynomial such that P(1)=1, P(2)=...

    Text Solution

    |

  7. Given f(x) is a polynomial function of x, satisfying f(x). f(y)=f(x)+f...

    Text Solution

    |

  8. A function f(x) is such that f(x)=0 has 8 distinct real roots and f(x)...

    Text Solution

    |

  9. Let: f:{x1, x2, x3 }rarr {1,2,3,4, .., 13} be a function, The number o...

    Text Solution

    |

  10. The number of integral values of x satisfying the equation sgn([15/(1+...

    Text Solution

    |

  11. Let f(x)=(1-c x)/(1-c^2) . If f(x) gt 0 for évery x greater than zero....

    Text Solution

    |

  12. Let f:[0, oo) rarr [0, oo) be a function defined by f(x)=x^2+(k^2-3 k+...

    Text Solution

    |

  13. Let g:[-2,2] rarr R, where g(x)=x^(2015)+sgn(x)+[(x^2+1)/p] be an odd ...

    Text Solution

    |

  14. Let the function f: R rarr R be defined as f(x)=min .(x+2,4-2 x, 1+4 x...

    Text Solution

    |

  15. Let f: R rarr R be defined as f(x)=2 x^3+7 x-5 and g(x)=f^-1(x). If g^...

    Text Solution

    |

  16. Let f: R rarr R be defined as f(x)={[2 x+3, x le 1 ] ,[p^2 x+1, x gt ...

    Text Solution

    |

  17. Let g: R rarr {3/2} rarr R-{3/2} be defined as g(x)=(3 x+5)/(2 x-3).If...

    Text Solution

    |

  18. If f(x)= 3 sqrt((9)/(log2 (3-2x) - 1) then the value of ' a ' which sa...

    Text Solution

    |

  19. Let f: R rarr R defined by f(x)=100^gn x+e^|x|, where sgn (x) denotes ...

    Text Solution

    |

  20. Consider the function f(x)=sqrt(2-x)+sqrt(1+x). If d denotes the numb...

    Text Solution

    |

  21. If f(x, y)=(max (x, y))^(min (x, y)) and g(x, y)=max (x, y) -min (x, y...

    Text Solution

    |