First slide

Theory of expressions

Question

The number of values of $k$ for which the equation $x^{2} - 2 x + k = 0$ has two distinct roots lying in the interval (0, 1) is

Moderate

A

0
B

1
C

2
D

infinitely many

Solution

Let the two distinct roots lying between 0 and 1 be
$α, β$ such that $α < β .$
As $f (x) = x^{2} - 2 x + k$ is a differentiable and $f (α) = f (β) = 0$ , therefore by the Roll’s theorem there exist $γ \in (α, β) \subset (0, 1)$ such that $f^{'} (γ) = 0$
$\Rightarrow 2 γ - 2 = 0 \Rightarrow γ = 1$ .Not possible.

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