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<-sqrt(3)` (c)`-sqrt(3)<=a<=` `sqrt(3)` (d) `non eoft h e s e`

Options

A. `a gt 0 `
B. ` a le sqrt(3) `
C. ` |a| le sqrt(3) `
D. none of these

Answer

C

Explanation

We have,
` y= x^(3)-ax^(2)+x+1 " "...(i) `
` rArr (dy)/(dx) =3x^(2)-2ax+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.
` therefore (dy)/(dx) ge 0 " for all "(x,y)` lying on the curve (i).
` rArr 3x^(2)-2ax+1 ge 0 " for all " x`
` rArr 4a^(2)-12 le 0 rArr a^(2)-3 le 0 rArr - sqrt(3) le a le sqrt(3) rArr |a| le sqrt(3) `