Question

If at each point of the curve $y=x^3-a{x}^2+x+1$, 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. $-\sqrt{3}<a<\sqrt{3}$
• D. None of these

Answer 1

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

$\frac{dy}{dx}=3x^2-2ax+1$

$tan\theta >0$ if $\theta$ is an acute angle.

Hence,

$3x^2-2ax+1>0$

which is true iff $a>0$ and $D<0$

$(2a{)}^2-4(3\left)\right(1)<0$

$a^2-3<0$

or, $-\sqrt{3}<a<\sqrt{3}$