Home
Class 12
MATHS
Let f(x+y)=f(x)+f(y) for all real x,y an...

Let `f(x+y)=f(x)+f(y)` for all real `x,y and f'(0)` exists. Prove that `f'(x) = f'(0)` for all `x in R and 2f(x) = xf (2).`

Promotional Banner

Topper's Solved these Questions

  • Differentiation

    A DAS GUPTA|Exercise Exercise|98 Videos

  • Differential Equation of the First Order

    A DAS GUPTA|Exercise Exercise|64 Videos

  • Elementary Probability

    A DAS GUPTA|Exercise Exercise|137 Videos

Similar Questions

Explore conceptually related problems

If f(x+y)=f(x)f(y) for all real x and y, f(6)=3 and f'(0)=10 , then f'(6) is

Let f(x+2y)=f(x)(f(y))^(2) for all x,y and f(0)=1. If f is derivable at x=0 then f'(x)=

Let f((x+y)/(2))=(f(x)+f(y))/(2) for all real x and y If f'(0) exists and equals -1 and f(0)=1, then find f(2)

A function f:R rarr R satisfies the relation f((x+y)/(3))=(1)/(3)|f(x)+f(y)+f(0)| for all x,y in R. If f'(0) exists,prove that f'(x) exists for all x,in R

f(x+y)=f(x)*f(y) for all x,y anf f(5)=2,f'(0)=3 then f(5) will be

Let f((x+y)/(2))=(f(x)+f(y))/(2) for all real x and y.If f(0) exists and equals-1and f(0)=1 find f(2)

If f(x+y)=f(x).f(y) for all real x,y and f(0)!=0, then the function g(x)=(f(x))/(1+{f(x)}^(2)) is:

A DAS GUPTA-Differentiation-Exercise
  1. A function f is defined such that for all real x, y (a) f(x+y)=f(x).f(...

    Text Solution

    |

  2. A function f is defined such that for all real x, y (a) f(x+y)=f(x).f(...

    Text Solution

    |

  3. Let f(x+y)=f(x)+f(y) for all real x,y and f'(0) exists. Prove that f'(...

    Text Solution

    |

  4. Let f((x+y)/(2))=1/2 |f(x) +f(y)|for all real x and y, if f '(0) exist...

    Text Solution

    |

  5. A function f(x) is so defined that for all real x,{f(x)}^n = f(nx). Pr...

    Text Solution

    |

  6. Prove that the derivative of (a) an odd function is an even function;

    Text Solution

    |

  7. Prove that the derivative of a periodic function of period T is a peri...

    Text Solution

    |

  8. Let f(x) be a function satisfying the condition f(-x) = f(x) for all r...

    Text Solution

    |

  9. A function f:R->R satisfies the relation f((x+y)/3)=1/3|f(x)+f(y)+f(0)...

    Text Solution

    |

  10. Find the sum of series sum(r=1)^nr.x^(r-1), using calculus.

    Text Solution

    |

  11. Differential coefficient of log2(log2x)w.r.t x is

    Text Solution

    |

  12. If f(x)=logx(logex) then f'(e)=

    Text Solution

    |

  13. If x=a (cos t +log (tan ((t)/(2)) )) ,y =a sin t ,then (dy)/(dx) =

    Text Solution

    |

  14. Let f'(x) = sin(x^2) and y = f(x^2 +1) then dy/dx at x=1 is

    Text Solution

    |

  15. If x=cost , y=loget then at t=pi/2, (d^2y)/(dx^2)+((dy)/(dx))^2=

    Text Solution

    |

  16. If f'(x)=sin x^2 and y=f(x^2+1) then dy/dx=

    Text Solution

    |

  17. If x=cost , y=loget then at t=pi/2, (d^2y)/(dx^2)+((dy)/(dx))^2=

    Text Solution

    |

  18. If tany = (2t)/(1-t^2) and sin x = (2t)/(1+t^2) then (dy)/(dx)=

    Text Solution

    |

  19. If xe^(xy)=y+sin^2x then at x=0 (dy)/dx=

    Text Solution

    |

  20. If x^2y +y^3=2, the value of (dy)/(dx) at the point (1,1) is

    Text Solution

    |