Question

for all x belongs to R, then, Lambda lies in which interval?

Answer 1

Answer:

for all x belongs to R   $\lambda\geq\frac{4}{61}$        ( i.e )    $\lambda \in [\frac{4}{61} , \infty)$

Step-by-step explanation:

given:

$\lambda x^2-9 \lambda x + 5 \lambda + 1 > 0$

This equation is quadratic on ' x ' ( form : $ax^2+bx+c$ )

a = $\lambda$      b = $-9\lambda$    c = $5\lambda + 1$

we know that,

Δ = $b^2-4ac$

for x ∈ R   Δ ≥ 0.

$(-9\lambda)^2 - 4(\lambda)(5\lambda + 1) \geq 0$

$81(\lambda)^2 - 20 (\lambda)^2-4\lambda\geq0$

$61(\lambda)^2-4\lambda\geq0$

$61(\lambda)^2\geq4\lambda$

$61\lambda\geq4$

$\lambda\geq\frac{4}{61}$

Thus you got the answer ! lucky !

Hope it helps !