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.
$$a \leqslant \sqrt 3 $$
C.
$$ - \sqrt 3 \leqslant a \leqslant \sqrt 3 $$
D.
none of these
Answer :
$$ - \sqrt 3 \leqslant a \leqslant \sqrt 3 $$
Solution :
$$\eqalign{
& \frac{{dy}}{{dx}} = 3{x^2} - 2ax + 1 \cr
& {\text{From the question,}}\,\,\,\,\frac{{dy}}{{dx}} \geqslant 0 \cr
& \Rightarrow 3{x^2} - 2ax + 1 \geqslant 0\,\,{\text{for all }}x \cr
& \therefore D \leqslant 0\,\,\,\,\,{\text{or,}}\,\,4{a^2} - 12 \leqslant 0\,\,\,\,\,\,\,\, \Rightarrow - \sqrt 3 \leqslant a \leqslant \sqrt 3 \cr} $$