The number of values of k for which the equation x3−3x+k=0 has two distinct roots lying in the interval (0,1) are

# The number of values of k for which the equation ${x}^{3}-3x+k=0$ has two distinct roots lying in the interval $\left(0,1\right)$ are

1. A
2. B
3. C
4. D

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### Solution:

Let there be a value of k for which ${x}^{3}-3x+k=0$

has two distinct roots between 0 and 1.

Let $a,b$ be two distinct roots of ${x}^{3}-3x+k=0$ lying between 0 and 1 such that $a

Let $f\left(x\right)={x}^{3}-3x+k\text{.}$Then

Between any two roots of a polynomial  there exists at least one roots of its derivative  Therefore,has at least one root between a and b. But, $f\text{'}\left(x\right)=0$ has two roots equal to± 1 which do not lie between a and b. Hence, $f\left(x\right)=0$ has no real roots lying between 0 and 1 for any value of

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