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

1. $a \leqslant \sqrt 3$
2. $a > 0$
3. $- \sqrt 3 < a \leqslant \sqrt 3$
4. None of these

Answer 1

Hint : We are given an equation of the curve. Since the tangent is inclined at an acute angle, this means that $\theta < {90^\circ }$. Next, we will apply a derivative on both sides with respect to x and so, the slope of the tangent $\dfrac{{dy}}{{dx}} = \tan \theta > 0$ . Next, we will find the value of $\Delta$ which is less than zero. We know that, $\Delta = {b^2} - 4ac$ and using this, we will find the final output.

Complete step-by-step answer :
Given that, the tangent is inclined at an acute angle, $\theta < {90^\circ }$
Also, given that, the curve is $y = {x^3} - a{x^2} + x + 1$.
Applying derivate on both the sides with respect to x, we will get,
$\Rightarrow \dfrac{{dy}}{{dx}} = 3{x^2} - 2ax + 1$
As we know that, slope of the tangent is $\dfrac{{dy}}{{dx}} = \tan \theta$
Since, $\theta < {90^\circ }$ then, $\tan \theta > 0$.
$\Rightarrow \dfrac{{dy}}{{dx}} > 0$
$\Rightarrow 3{x^2} - 2ax + 1 > 0$ for all x which belong to the real number.
This will be true if and only if $a > 0$ and $\Delta < 0$
Now, we will find this value,
$\Delta < 0$
$\Rightarrow {( - 2a)^2} - 4(3)(1) < 0$ $(\because \Delta = {b^2} - 4ac)$
$\Rightarrow 4{a^2} - 12 < 0$
$\Rightarrow 4{a^2} < 12$
$\Rightarrow {a^2} < 3$
$\therefore - \sqrt 3 < a < \sqrt 3$
Hence, for each point of the curve, the tangent is inclined at the acute angle with positive direction of x-axis then the value is $- \sqrt 3 < a < \sqrt 3$ .
So, the correct answer is “Option B”.

Note : A tangent is a line which represents the slope of a curve at that point. A slope of a line is calculated by dividing the change in height by the change in horizontal distance. Tangent is a straight line (or smooth curve) that touches a given curve at one point and at that point the slope of the curve is equal to that of the tangent.