Question

If tangent at each point to the curve $y=2x^3+a{x}^2+6x+5$ is inclined at an acute angle, then the number of integral values of $a$ is

Answer 1

Given curve: $y=2x^3+a{x}^2+6x+5$
Differentianting both sides w.r.t. $x$
$\Rightarrow \frac{dy}{dx}=6x^2+2ax+6$
as tangent inclined at an acute angle
so, $\theta \in (0,\frac{\pi }{2})$
$\Rightarrow 6x^2+2ax+6>0$ for all $x.$
$\Rightarrow D<0$
$\Rightarrow 4a^2-144<0$
$\Rightarrow (a+6)(a-6)<0$
$\Rightarrow -6<a<6$
so, number of integral values of $a$ are $11.$