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

$|a| \leq \sqrt{3}$

We have,

$y=x^3-a x^2+x+1$            .......(i)

$\Rightarrow \frac{d y}{d x}=3 x^2-2 a x+1$

It is given that at each point on the curve (i), the tangent is inclined at an acute angle with the positive direction of x-axis.

∴  $\frac{d y}{d x} \geq 0$ for all (x, y) lying on the curve (i).

$\Rightarrow 3 x^2-2 a x+1 \geq 0$ for all x

$\Rightarrow 4 a^2-12 \leq 0 \Rightarrow a^2-3 \leq 0 \Rightarrow-\sqrt{3} \leq a \leq \sqrt{3} \Rightarrow|a| \leq \sqrt{3}$