Question

The number of values of $\lambda$ for with the equation $x^2-3x\lambda =0$ has two distinct roots lying in the interval $(0,1)$ is

Answer 1

Consider the given function

$f\left(x\right)=x^2-3x\lambda =0$

Differentiating with respect to x and we get,

$f^{\prime }\left(x\right)=2x-3\lambda =0$ …… (1)

Put $x=(0,1)$ and we get,

$f^{\prime }\left(0\right)=2\times 0-3\lambda =0$

$\lambda =0$

$f^{\prime }\left(1\right)=2\times 1-3\lambda =0$

$2-3\lambda =0$

$\lambda =\frac{2}{3}$

Now,$0<\frac{2}{3}<1$

Hence, this is the answer