Question

If the tangent at each point of the curve $y=\frac{2}{3}{x}^3-2a{x}^2+2x+5$ makes an acute angle with positive direction of X-axis then

• A. $a\ge 1$
• B. $-1\le a\le 1$
• C. $a\le -1$
• D. None of these

Answer 1

The correct option is B $-1\le a\le 1$

On differentiating, We get

$\frac{dy}{dx}=2x^2-4ax+2$

If Tangent makes acute angle with +ve x axis, then

$\frac{dy}{dx}\ge 0$ For all x

$x^2-2ax+1\ge 0\Rightarrow 4a^2-4\le 0\Rightarrow a^2\le 1-1\le a\le 1$