Statement-1 : There is no value ofb k for which the equaiton `x^(3)-3x+k=0` has two distinct roots between 0 and 1 .
Statement-2: `x gt sin x "for all" x gt0`

Options

A. Statement-1 is True, Statement-2 is Ture, Statement-2 is a correct explanation for statement-1
B. Statement-1 is True, Statement-2 is Ture, Statement-2 is not a correct explanation for statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 True.

Answer

B

Explanation

Let there be two distinct roots of `f(x)=x^(3)-3x+k=0` in the interval [0,1] then, f'(x)=0 i.e., `3x^(2)-3=0` must have a root between 0 and 1 . But `3x^(2)-2=0` does not have a root between 0 and 1 . This contradicts the algebraci interpretation of Rolle's Theforem, So,f(x)=0 does not have two distinct roots between 0 and 1.
Hence, statement-1n is ture.
Let `g(x)=x -sin x and xgt0` by any real, number.
Apllying Largrang'es mean value theorem on [0,x] we have `g'(c)(g(x)-g(0))/(x-0) "for some" c in (0,x)`
`rArr 1-cos c=(x-sin x)/(x)" " [ :. 1 -cos gt0]`
`rArr (x-sin x)/(x)gt0 `
`rArr x-sin gt0 " " [ :. x gt0]`
`rArr x sin x`
So, statement-2 is also, true. But it is not a correct explanation for statement-1