Question

The range of $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$ $\forall xϵ$ R is a subset of $[0,\frac{\pi }{2})$ then the range of a is -

• A. $[-\sqrt{3},\frac{1}{4}]$
• B. $(\frac{-\pi }{2},\frac{\pi }{2})$
• C. $[-\sqrt{3},-1]$
• D. $[\frac{1}{4},\infty )$

Answer 1

The correct option is D $[\frac{1}{4},\infty )$

$ta{n}^{-1}(x^2+x+a)\ge 0\Rightarrow x^2+x+a\ge 0$
$\Rightarrow D\le 0\Rightarrow 1-4a\le 0\Rightarrow a\ge \frac{1}{4}$ ; D is discriminant of quadratic equation.
$\Rightarrow aϵ[\frac{1}{4},\infty )$