Question

If at each point of the curve $y=x^3-a{x}^2+x+1$, the tangent is inclined at an acute angle with the positive direction of the $x$-axis, then

• A. $a>0$
• B. $a\le \sqrt{3}$
• C. $-\sqrt{3}\le a\le \sqrt{3}$
• D. none of these

Answer 1

The correct option is C $-\sqrt{3}\le a\le \sqrt{3}$

$y=x^3-a{x}^2+x+1$
$\frac{dy}{dx}=3x^2-2ax+1$
Since, the tangent is inclined at an acute angle with the positive direction of x-axis.
$\frac{dy}{dx}>0$
$\Rightarrow 3x^2-2ax+1>0$
$\Rightarrow D<0$
$\Rightarrow 4(a^2-3)<0$
$\Rightarrow (a-\sqrt{3})(a+\sqrt{3})<0$
$\Rightarrow -\sqrt{3}<a<\sqrt{3}$