Home
Class 12
MATHS
Find a function g:R rarr R continous in ...

Find a function `g:R rarr R` continous in `[0,infty]` satisfying g(0) =1 and `1/2 int_0^x g^2(t) dt=1/x(int_0^xg(t)dt)^2`)

Text Solution

Verified by Experts

g(x)=`cx^(1+sqrt2)`
Promotional Banner

Topper's Solved these Questions

  • CONTINUITY & DIFFERENTIABILITY

    PATHFINDER|Exercise QUESTION BANK|13 Videos

  • DETERMINATES

    PATHFINDER|Exercise QUESTION BANK|21 Videos

Similar Questions

Explore conceptually related problems

Let f : R rarr R be a continuous function which satisfies f(x) = int_0^x f(t) dt . Then the value of f(log_e 5) is

Prove that int_0^x e^(x t-t^2) dt=e^(x^(2)/4)int_0^x e^-(t^(2)/4)dt

Let f be an odd continous function which is periodic with priod 2 if g(x) = int_0^x f(t) dt then

Let f:[a,b]rarr[1,infty) be a continuous function and lt g: RrarrR be defined as g(x)={(0, "if x" le a),(int_a^x f(t)dt, "if a"lexleb),(int_a^b f(t)dt, "if x"gtb):} Then

Let f:[a,b]rarr[1,infty) be a continuous function and lt g: RrarrR be defined as g(x)={(0, "if x" lt a),(int_a^x f(t)dt, "if a"lexleb),(int_a^b f(t)dt, "if x"gtb):} Then a) g(x) is continuous but not differentiable at a b) g(x) is differentiable on R c) g(x) is continuous but nut differentiable at b d) g(x) is continuous and differentiable at either a or b but not both.

If a continous function f satisfies int_0^f(x) t^2 dt=x^2(1+x) for all xge0 then f(2) is equal to

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an odd function, then

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b If f(x) is an even function, then

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0 , and int_(b)^(2k)g(t)dt is independent of b Least positive value fo c if c,k,b are n A.P. is

f: R to R is a continuous and f(x)=int_(0)^(x)f(t)dt then f(log_(e)5)= ?

PATHFINDER-DEFINITE INTEGRATION-QUESTION BANK
  1. If Un=int0^pi (1-cosnx)/(1-cosx) dx where n is positive integer or zer...

    Text Solution

    |

  2. Prove that int0^x e^(xt)e^(-t^2)dt =e^(x^2/4) int0^xe^(-t^2/4)dt

    Text Solution

    |

  3. Find a function g:R rarr R continous in [0,infty] satisfying g(0) =1 a...

    Text Solution

    |

  4. Evaluate underset(nrarrinfty)lim 1/n overset(4n)underset(r=1)sum r/(sq...

    Text Solution

    |

  5. If In=int0^1 x^ntan^(-1)x dx then prove that (n+1)In +(n-1)I(n-2)=(pi/...

    Text Solution

    |

  6. If f(x)=x+int0^1(xy^2+x^2y) dy find f(x)

    Text Solution

    |

  7. The total number of distinct x in[0,1] for which int0^x t^2/(1+t^4) dt...

    Text Solution

    |

  8. The value of int(-pi/2)^(pi/2) (x^2cosx)/(1+e^x) dx is equal to

    Text Solution

    |

  9. underset(nrarrinfty)lim((n+1)(n+2)....3n)/(n^(2n)))^(1/n) is equal to

    Text Solution

    |

  10. int0^5 ((1/x)-1) dx= ?

    Text Solution

    |

  11. The value of underset(nrarrinfty)lim {(sqrt(n+1) +sqrt(n+2) +......+sq...

    Text Solution

    |

  12. if [x] denotes the greatest integer less than or equal to x then integ...

    Text Solution

    |

  13. (phi)t={1, for 0lelt1 0 otherwise then int(-3000)^(3000) (underset(r...

    Text Solution

    |

  14. Let f:RrarrR be a function defined by f(x)={([x],x,le,2),(0,x,gt,2):} ...

    Text Solution

    |

  15. If alpha=int0^1(e^(9x+3tan^-1x))((12+9x^2)/(1+x^2))dx where tan^-1x t...

    Text Solution

    |

  16. Let f: R rarr R be a continuous odd function, which vanishes exactly a...

    Text Solution

    |

  17. Let f(x)=7tan^8x+7tan^6x-3tan^4x-3tan^2x for all x in (-pi/2, pi/2). T...

    Text Solution

    |

  18. If f(x) + 2f(1 - x) =x^2 + 2, forall x in R, then f(x) is

    Text Solution

    |

  19. The option(s) with the values of a and L that satisfy the following eq...

    Text Solution

    |

  20. Let F:RrarrR be a thrice differentiable function. Suppose that F(1)=0,...

    Text Solution

    |