If the tangent at each point of the curve $y=\frac{2}{3} x^3-2 a x^2+2 x+5$ makes an acute angle with the positive direction of x-axis, then
Answer & explanation
Correct answer: option 2
It is given that the tangent at each point of the curve
$y=\frac{2}{3} x^3-2 a x^2+2 x+5$
makes an acute angle with the positive direction of x-axis.
∴ $\frac{d y}{d x} \geq 0$ for all x
$\Rightarrow 2 x^2-4 a x+2 \geq 0$ for all x
$\Rightarrow x^2-2 a x+1 \geq 0$ for all x
$\Rightarrow 4 a^2-4 \leq 0 \Rightarrow a^2-1 \leq 0 \Rightarrow-1 \leq a \leq 1$