Home
Class 12
MATHS
Let f be a real-valued function defined ...

Let f be a real-valued function defined on the interval `(0,oo)` by `f(x) = lnx+int_(0)^(x)sqrt(1+sint)dt`. Then which of the following statement(s) is (are) true ?

A

`f"(x)` exist for all `x in (0,oo)`

B

`f'(x)` exists for all `x in (0,oo)` and `f'` is continuous on `(0,oo)` but not differentiable on `(0,oo)`

C

there exists `alpha gt 1` such that `|f'(x)| lt |f(x)|` for all `x in (alpha, oo)`

D

there exists `beta gt 0` such that `|f(x)| + |f'(x)| le beta` for all `x in (0,oo)`

Text Solution

Verified by Experts

The correct Answer is:
B, C
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION & ITS APPLICATION

    RESONANCE|Exercise Exercise 3 Part - II|25 Videos

  • DEFINITE INTEGRATION & ITS APPLICATION

    RESONANCE|Exercise High Level Problem|37 Videos

  • DEFINITE INTEGRATION & ITS APPLICATION

    RESONANCE|Exercise Exercise 2 Part - IV|9 Videos

  • COMBINATORICS

    RESONANCE|Exercise Exercise-2 (Part-II: Previously Asked Question of RMO)|8 Videos

  • DPP

    RESONANCE|Exercise QUESTION|665 Videos

Similar Questions

Explore conceptually related problems

Let f be a real-valued function defined on interval (0,oo),by f(x)=lnx+int_0^xsqrt(1+sint).dt. Then which of the following statement(s) is (are) true? (A). f"(x) exists for all in (0,oo)." " (B). f'(x) exists for all x in (0,oo) and f' is continuous on (0,oo), but not differentiable on (0,oo)." " (C). there exists alpha>1 such that |f'(x)|<|f(x)| for all x in (alpha,oo)." " (D). there exists beta>1 such that |f(x)|+|f'(x)|<=beta for all x in (0,oo).

Let f(x) be a differentiable function in the interval (0,2) then the value of int_(0)^(2)f(x) is

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

Let f(x) be a differentiable function in the interval (0,2) , then the value of int_0^2 f(x) dx is :

Let f:[0,oo)rarrR be a continuous function such that f(x)=1-2x+int_(0)^(x)e^(x-t)f(t)dt" for all "x in [0, oo). Then, which of the following statements(s) is (are)) TRUE?

Let f be a real valued function such that for any real x f(15+x)=f(15-x)and f(30+x)=-f(30-x) Then which of the following statements is true ?

RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -Exercise 3 Part - I
  1. The value of lim(xrarr0)(1)/(x^(3))int(x)^(0) (tln(1+t))/(t^(4)+4) dt ...

    Text Solution

    |

  2. The value(s) of int(0)^(1) (x^(4)(1-x)^(4))/(1+x^(2))dx is are

    Text Solution

    |

  3. Let f be a real-valued function defined on the interval (0,oo) by f(x)...

    Text Solution

    |

  4. For any real number, let [x] denotes the largest integer less than or ...

    Text Solution

    |

  5. Let f be a real-valued function defined on the interval (-1,1) such th...

    Text Solution

    |

  6. Consider the polynomial f(x)= 1+2x+3x^2+4x^3. Let s be the sum of all ...

    Text Solution

    |

  7. Consider the polynomial f(x)= 1+2x+3x^2+4x^3. Let s be the sum of all ...

    Text Solution

    |

  8. Consider the polynomial f(x)= 1+2x+3x^2+4x^3. Let s be the sum of all ...

    Text Solution

    |

  9. The value of int(sqrt(ln2))^(ln3)(xsinx^(2))/(sinx^(2)+sin(ln6-x^(2)))...

    Text Solution

    |

  10. Let the straight line x = b divide the area enclosed by y = (1-x)^(2),...

    Text Solution

    |

  11. Let f : [-1,2]rarr[0,oo) be a continuous function such that f(x) = f(1...

    Text Solution

    |

  12. Let S be the area of the region enclosed by y=e^-x^2,y=0,x=0,a n dx=1....

    Text Solution

    |

  13. The value of the integral int(-pi//2)^(pi//2) (x^(2)+ln'(pi+x)/(x-a)) ...

    Text Solution

    |

  14. Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = in...

    Text Solution

    |

  15. Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = in...

    Text Solution

    |

  16. If f(x) = int(0)^(x) e^(t^(2)) (t-2) (t-3) dt for all x in (0, oo) , t...

    Text Solution

    |

  17. The area enclosed by the curvesy= sinx+cosx and y = | cosx-sin x | ove...

    Text Solution

    |

  18. Let f : [1/2, 1] rarr R (the set of the all real numbers) be a positiv...

    Text Solution

    |

  19. For a in R (the set of all real numbers), a ne = - 1, lim(nrarroo)((...

    Text Solution

    |

  20. Let f:[a,b]to[1,oo) be a continuous function and let g:RtoR be defined...

    Text Solution

    |