Question

If $x^2+2ax+a<0,\forall x\in [1,2]$ then the values of ${}^{\prime }{a}^{\prime }$ lies in the interval

• A. $(-\infty ,-\frac{4}{5})$
• B. $(-\infty ,-\frac{1}{3})$
• C. $(-\infty ,-\frac{1}{3}]$
• D. $(-\infty ,-\frac{4}{5}]$

Answer 1

The correct option is A $(-\infty ,-\frac{4}{5})$

Let $f\left(x\right)=x^2+2ax+a$
Given:
$f\left(x\right)<0,\forall x\in [1,2]$
$\Rightarrow$ $1$ and $2$ lie between the roots of $f\left(x\right)=0$


Conditions:
$\left(i\right)f\left(1\right)<0\Rightarrow 1+3a<0\Rightarrow a<-\frac{1}{3}\dots \dots \left(1\right)$

$\left(iii\right)f\left(2\right)<0\Rightarrow 4+5a<0\Rightarrow a<-\frac{4}{5}\dots \dots \left(2\right)$

From $\left(1\right)\text{and}\left(2\right)$,
We get $a\in (-\infty ,-\frac{4}{5})$