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

The number of values of k for which the equation $x^{3} - 3 x + k = 0$ has two different roots lying in the interval (0, 1) is

Let $f (x) = x^{3} - 3 x + k .$ Assume, if possible, $a, b \in (0, 1)$ be the different roots of
f(x)

= 0. f(x) is continuous on [0, 1] and differentiable on (0, 1).

Also, f(a) = f(b) = 0

$∴$ By Rolle’s theorem, there exists $c \in (0, 1)$ such that

$f^{'} (c) = 0 \Rightarrow 3 c^{2} - 3 = 0 \Rightarrow c = \pm 1$

$∴$ No value of $k$ satisfies the condition

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