Home
Class 12
MATHS
Let f(x) be periodic and k be a positive...

Let `f(x)` be periodic and `k` be a positive real number such that `f(x+k)+f(x)=0fora l lx in Rdot` Prove that `f(x)` is periodic with period `2kdot`

Text Solution

Verified by Experts

We have `f(x+k)+f(x)=0 AA x in R`
or ` f(x+k)= -f(x) AA x in R`
Put `x=x+k.` Then,
or `f(x+2k)= -f(x+k) AA x in R`
or ` f(x+2k) = f(x), AA x in R " "[ "As "f(x+k)= -f(x)]`
which clearly shows that f(x) is periodic with period 2k.
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE PUBLICATION|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

If f(x+2a)=f(x-2a) ,then prove that f(x) is periodic

If f(x) satisfies the relation f(x)+f(x+4)=f(x+2)+f(x+6) for all x , then prove that f(x) is periodic and find its period.

If a , b are two fixed positive integers such that f(a+x)=b+[b^3+1-3b^2f(x)+3b{f(x)}^2-{f(x)}^3]^(1/3) for all real x , then prove that f(x) is periodic and find its period.

If a, b be two fixed positive integers such that f(a+x)=b+[b^3+1-3b^2 f(x)+3b f(x)^2-(f(x))^3]^(1/3) for all real x, then f(x) is a periodic function with period.

The function f(x)=1/2{|sin x|/cos x +sin x/|cos x|} is periodic with period

Prove that function f(x)=cos sqrt(x) is non-periodic.

Let the function ln f(x) is defined where f(x) exists for 2 geq x and k is fixed positive real numbers prove that if -k f(x) geq d/(dx) (x.f(x)) then f(x) geq Ax^(-k-1) where A is independent of x.

If f'(x)=(1-2sin^2x)/f(x), (f(x) ge 0, AA x in R and f(0)=1) then f(x) is a periodic function with the period

Let y=f(x) be drawn with f(0) =2 and for each real number a the line tangent to y = f(x) at (a,f(a)) has x-intercept (a-2) . If f(x) is of the form of k e^(px) then k/p has the value equal to

Let RR be the set of real numbers and f : RR to RR be defined by f(x)=sin x, then the range of f(x) is-

CENGAGE PUBLICATION-RELATIONS AND FUNCTIONS-All Questions
  1. Prove that f(x)gi v e nb yf(x+y)=f(x)+f(y)AAx in R is an odd function...

    Text Solution

    |

  2. If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that ...

    Text Solution

    |

  3. Let f(x) be periodic and k be a positive real number such that f(x+k)+...

    Text Solution

    |

  4. If f(x) satisfies the relation f(x)+f(x+4)=f(x+2)+f(x+6) for allx , th...

    Text Solution

    |

  5. An odd function is symmetric about the vertical line x=a ,(a >0),a n d...

    Text Solution

    |

  6. Check whether the function defined by f(x+lambda)=1+sqrt(2f(x)-f^2(x))...

    Text Solution

    |

  7. Draw the graph of y=-log(e)x.

    Text Solution

    |

  8. Draw the graph of y=|||x|-2|-3| by transforming the graph of y=|x|

    Text Solution

    |

  9. Consider the function f(x)={{:(2x+3",",xle1),(-x^(2)+6",",xgt1):} Th...

    Text Solution

    |

  10. Sketch the curve |y|=(x-1)(x-2)dot

    Text Solution

    |

  11. If f(x)=3x+1,g(x)=x^3+2, then (f+g)/(fg) (0) =?

    Text Solution

    |

  12. Find the range of f(x)=sin(cosx).

    Text Solution

    |

  13. Let g(x)=sqrt(x-2k),AA2klt=x<2(k+1), where k in integer. Check whethe...

    Text Solution

    |

  14. Let f(x)=x^2-2x ,x in R ,a n dg(x)=f(f(x)-1)+f(5-(x))dot Show that g(...

    Text Solution

    |

  15. If fa n dg are two distinct linear functions defined on R such that th...

    Text Solution

    |

  16. Let f(x)=(log)2(log)3(log)4(log)5(sin x+a^2)dot Find the set of values...

    Text Solution

    |

  17. If f is polynomial function satisfying 2+f(x)f(y)=f(x)+f(y)+f(x y)AAx ...

    Text Solution

    |

  18. Let f:XtoY be a function defined by f(x)=asin(x+pi/4)+bcosx+c. If f is...

    Text Solution

    |

  19. Let f: RtoR ,f(x)=(x-a)/((x-b)(x-c)),b > c If f is onto, then prove...

    Text Solution

    |

  20. If p and q are positive integers, f is a function defined for positive...

    Text Solution

    |