First slide

Theory of expressions

Question

If the equation $x^{3} - 3 x + a = 0$ has distinct roots between 0 and 1, then the value of a is

Moderate

A

2
B

$1 / 2$
C

3
D

none of these

Solution

Let $f (x) = x^{3} - 3 x + a$
If $f (x)$ has distinct roots between 0 and 1. Then,
by Rolle's theorem $f' (x) = 0$ has a root between 0 and 1.
But , $f^{'} (x) = 0 \Rightarrow 3 x^{2} - 3 = 0 \Rightarrow x = \pm 1$
Clearly, $f' (x) = 0$ does not have any root between 0 and 1. So, $f (x)$ does not have distinct roots between 0 and 1 for any value of $a .$

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