Home
Class 12
MATHS
Let f(x)=x^3+a x^2+b x+5sin^2x be an inc...

Let `f(x)=x^3+a x^2+b x+5sin^2x` be an increasing function on the set `Rdot` Then find the condition on `a` and `b`.

A

`a^2-3b-15 gt 0`

B

`a^2-3b+15 gt 0 `

C

`a^2-3b+15 lt 0 `

D

`a gt 0 and b lt 0 `

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)=x^3+ax^2+bx + 5 sin ^2x` increasing on R
`rArr (x) lt for all in R`
`rArr 3x^2a + (b-5)gt 0 for all x in R`
`rArr 3x^2+2ax(b-5) lt 0 rArr a^2-3b+15 lt 0 `
Promotional Banner

Topper's Solved these Questions

  • INCREASING AND DECREASING FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|8 Videos

  • INCREASING AND DECREASING FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|36 Videos

  • INCREASING AND DECREASING FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|18 Videos

  • HEIGHTS AND DISTANCES

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|45 Videos

  • INDEFINITE INTEGRALS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos

Similar Questions

Explore conceptually related problems

Let f(x)=x^3+kx^2+5x+4sin^2x be an increasing function on x in R. Then domain of k is

If f(x)=x^3+a x^2+b x+5sin^2x\ is a strictly increasing function on the set of real numbers then a and b must satisfy the relation:

Find the value(s) of a for which f(x)=x^3-a x is an increasing function on Rdot

Let f(x)=[b^(2)+(a-1)b+2]x-int(sin^(2)x+cos^(4)x)dx be an increasing function of x""inRandbinR , then " a " can take value(s)

Let f(x) = ax^3 + bx^2 + cx + d sin x . Find the condition that f(x) is always one-one function.

Prove that f(x)=a x+b , where a , b are constants and a >0 is an increasing function on Rdot

Show that f(x)=e^(2x) is increasing on Rdot

Show that the function f(x)=2x+3 is strictly increasing function on Rdot

Let f (x) = ax+cos 2x +sin x+ cos x is defined for AA x in R and a in R and is strictely increasing function. If the range of a is [(m)/(n),oo), then find the minimum vlaue of (m- n).

Let f be the function f(x)=cosx-(1-(x^2)/2)dot Then (a) f(x) is an increasing function in (0,oo) (b) f(x) is a decreasing function in (-oo,oo) (c) f(x) is an increasing function in (-oo,oo) (d) f(x) is a decreasing function in (-oo,0)

OBJECTIVE RD SHARMA ENGLISH-INCREASING AND DECREASING FUNCTIONS-Section I - Solved Mcqs
  1. For x gt 1 and y = lod x which one of the following is not true ?

    Text Solution

    |

  2. If the function f(x)=2x^2-k x+5 is increasing on [1,\ 2] , then k lies...

    Text Solution

    |

  3. Let f(x)=x^3+a x^2+b x+5sin^2x be an increasing function on the set Rd...

    Text Solution

    |

  4. If a ,\ b ,\ c are real numbers, then find the intervals in which f(...

    Text Solution

    |

  5. Let f(x) be the function given by f(x) = 3x^5-5x^3+21x+3sinx+cosx+5....

    Text Solution

    |

  6. If the function f(x)=2tanx+(2a+1)(log)e|secx|+(a-2)x is increasing on ...

    Text Solution

    |

  7. If f'(x^2-4x+3)gt0 " for all " x in (2,3) then f(sinx) is increasing ...

    Text Solution

    |

  8. Let f(x) = tan^-1 (g(x)), where g (x) is monotonically increasing for...

    Text Solution

    |

  9. Let f(x) =inte^x(x-1)(x-2)dx, then f(x) decrease in the interval

    Text Solution

    |

  10. Assertion Consider the following statements in S and R S: Both sinx a...

    Text Solution

    |

  11. The length of the longest interval, in which the function 3sin x-4 sin...

    Text Solution

    |

  12. If f(x)=x^3+4x^2+ax+5 is a monotonically decreasing function of x in t...

    Text Solution

    |

  13. Let f(x) = 2sin^3x-3sin^2+12sinx+5, 0lexlepi/2 Then f(x)is

    Text Solution

    |

  14. Let f'(x) gt0 and g'(x) lt 0 " for all " x in R Then

    Text Solution

    |

  15. If f(x)= (ab+b^2+1)x+oversetx underset0int(cos^4theta+sin^4theta)" d"t...

    Text Solution

    |

  16. If f(x)=(P^(2)-1)/(P^(2)+1)x^(3)-3x+log2 is a decreasing function of ...

    Text Solution

    |

  17. If g(x) is a continous function at x=a such that g(a)gt0 and f'(x)= g(...

    Text Solution

    |

  18. Let g(x)=2f(x/2)+f(2-x)a n df^(x)<0AAx in (0,2)dot Then g(x) increases...

    Text Solution

    |

  19. The set of all values of a for which the function f(x)=((a^2-1)/3)x...

    Text Solution

    |

  20. Let f(x)={{:(xe^(ax) "," x le 0 ),(x+ax^2-x^3 "," x gt 0 ):},where is ...

    Text Solution

    |