Question

At each point of curve $y=x^3-3a{x}^2+6x+b,$ its tangent is inclined at an acute angle, then which of the following is/are true

• A. $a\in (-\sqrt{2},\sqrt{2})$
• B. $a\in R$
• C. $b\in (-\sqrt{2},\sqrt{2})$
• D. $b\in R$

Answer 1

Answer: (A), (D)

$b\in R$

$\frac{dy}{dx}=3x^2-6ax+6>0$
$\Rightarrow D<0$
$\Rightarrow 36a^2-4\cdot 3\cdot 6<0$
$\Rightarrow 6a^2-4\cdot 3<0$
$\Rightarrow a^2<2$
$\Rightarrow a\in (-\sqrt{2},\sqrt{2})$ and $b\in R$