If the tangent at each point of the curve
`y=(2)/(3) x^(3)-2ax^(2)+2x+5 `
makes an acute angle with the positive direction of x-axis, then

Options

A. ` a ge 1 `
B. `-1 le a le 1 `
C. ` a le -1 `
D. none of these

Answer

B

Explanation

It is given that the tangent at each point of the curve
` y=(2)/(3)x^(3)-2ax^(2)+2x+5 `
makes an acute angle with the positive direction of x-axis.
` therefore (dy)/(dx) ge 0` for all x
` rArr 2x^(2)-4ax+2 ge 0 ` for all x
`rArr x^(2)-2ax+1 ge 0 ` for all x
` rArr 4a^(2)-4 le 0 rArr a^(2)-1 le 0 rArr -1 le a le 1. `