Home
Class 12
MATHS
Statement 1: Let f: RvecR be a real-valu...

Statement 1: Let `f: RvecR` be a real-valued function `AAx ,y in R` such that `|f(x)-f(y)|<=|x-y|^3` . Then `f(x)` is a constant function. Statement 2: If the derivative of the function w.r.t. `x` is zero, then function is constant.

Text Solution

Verified by Experts

`"We have "|f(x)-f(y)|le|x-y|^(3),xney`
`therefore" "|(f(x)-f(y))/(x-y)|le|x-y|^(2)`
`rArr" "underset(yrarrx)lim|(f(x)-f(y))/(x-y)|leunderset(yrarrx)lim|x-y|^(2)`
`rArr" "|underset(yrarrx)lim(f(x)-f(y))/(x-y)|le|underset(yrarrx)lim(x-y)^(2)|`
`rArr" "|f'(x)|le0`
`rArr" "|f'(x)|=0" "(because|f'(x)|ge0)`
`therefore" "f'(x)=0`
`rArr" "f(x)=c" (constant)"`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE|Exercise Exercise 3.2|38 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Exercise 3.3|8 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Numerical Value Type|3 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Question Bank|25 Videos

  • DOT PRODUCT

    CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

Statement 1: Let f:R rarr R be a real-valued function AA x,y in R such that |f(x)-f(y)|<=|x-y|^(3) .Then f(x) is a constant function.Statement 2: If the derivative of the function w.r.t.x is zero,then function is constant.

Let f(x) be a real valued function not identically zero such that f(x+y^(2n+1))=f(x)+(f(y))^(2n+1), n epsilon N and x, y epsilo R . If f'(0)ge0 , then f'(6) is

Let f be a real valued function satisfying f(x+y)=f(x)f(y) for all x, y in R such that f(1)=2 . Then , sum_(k=1)^(n) f(k)=

Let f be a real valued function satisfying f(x+y)=f(x)f(y) for all x, y in R such that f(1)=2 . If sum_(k=1)^(n)f(a+k)=16(2^(n)-1) , then a=

Let f(x) be a real valued function such that f(0)=1/2 and f(x+y)=f(x)f(a-y)+f(y)f(a-x), forall x,y in R , then for some real a,

Let f(x) be real valued and differentiable function on R such that f(x+y)=(f(x)+f(y))/(1-f(x)f(y))f(0) is equals a.1 b.0 c.-1 d.none of these

Let f(x) be real valued and differentiable function on R such that f(x+y)=(f(x)+f(y))/(1-f(x)*f(y))f(x) is Odd function Even function Odd and even function simultaneously Neither even nor odd

Let f(x) be a real valued function satisfying the relation f((x)/(y))=f(x)-f(y) and lim_(x rarr0)(f(1+x))/(x)=3. The area bounded by the curve y=f(x), y-axis and the line y=3 is equal to