Home
Class 12
MATHS
If a continuous function of defined on t...

If a continuous function of defined on the real line R, assumes positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation `f(x)=0` has a root in R. Considetr `f(x)=ke^(x)-x` for all real x where k is real constant.
For `k gt 0,` the set of all values of k for which `ke^(x)-x=0` has two distinct, roots, is

A

`(0,(1)/(e))`

B

`((1)/(e),1)`

C

`((1)/(e),oo)`

D

`(0,1)`

Text Solution

Verified by Experts

The correct Answer is:
A
For two distinct roots, `1+ ln k lt 0(k gt0)`
`ln k lt-1impliesklt1/e`
Hence,` k in (0,(1)/(e))`
Promotional Banner

Similar Questions

Explore conceptually related problems

If a continous function of defined on the real line R, assumes positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. The positive value of k for which ke^(x)-x=0 has only root is

If a continuous function f defined on the real line R assume positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative, then the equation f(x)=0 has a root in R. Consider f(x)= ke^(x)-x , for all real x where k is a real constant. The line y=x meets y=ke^(x) for k le 0 at

The true set of real values of x for which the function f(x)=xlnx-x+1 is positive is

Find the maximum and the minimum values, if any, of the function f given by f(x) = x^(2) ,x in R .

Find the maximum and minimum values of f , if any, of the function given by f(x) = |x|, x in R.

consider the function f(X) =x+cosx -a values of a for which f(X) =0 has exactly one negative root are

If the equation 4x^(3)+5x+k=0(k in R) has a negative real root then

consider the function f(X) =x+cosx -a values of a which f(X) =0 has exactly one positive root are

Find local minimum value of the function f given by f (x) = 3 + |x|, x in R.

H-7.Let'f' be a real valued function defined for all real numbers x such that for some positive constant 'a' theequation f(x+a)= 2 1 ​ + f(x)−(f(x)) 2 holds for all x. Prove that the function f is periodic.