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(i) The positive value of k for which ke...

(i) The positive value of k for which `ke^x-x=0` has only one root is

A

`(1)/(e )`

B

1

C

e

D

log 2

Text Solution

Verified by Experts

The correct Answer is:
A
Consider the function `f(x)=ke^(x)-x`.
We have,
`f'(x)=ke^(x)-1`
`:. F'(x)=0 rArrke^(x)=1 rArr e^(x)=(1)/(k)rArrx=-logk`.
The signs of f' (x) for different values of x are as shown below.

So, f(x) attains its minimum value at x=-logk
Also, the minimum value of f(x) is f(-logk)=1+logk.
Thus, f(x)=0 will have exactly one root if its minimum value 1+log k is zero.
i.e. `1+log k=0 rArr k=1//e`
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OBJECTIVE RD SHARMA ENGLISH-MISCELLANEOUS EQUATIONS AND INEQUATIONS -Chapter Test
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  14. If the sum of the greatest integer less than or equal to x and the lea...

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