Question

If slope of tangent at atleast one point to the curve $y=x^3+3a{x}^2+3x+7$ is negative, then

• A. $a\in (-\infty ,0)$
• B. $a\in (-\infty ,-1)$
• C. $a\in (-1,1)$
• D. $a\in (1,\infty )$

Answer 1

Answer: (B), (D)

$a\in (1,\infty )$

Given curve: $y=x^3+3a{x}^2+3x+7$
Differentiating both sides w.r.t. $x$
$\Rightarrow y^{\prime }=3x^2+6ax+3$
Since, $y^{\prime }<0$ (for atleast one point)
$\therefore 3x^2+6ax+3<0$
$\Rightarrow D>0$
$\Rightarrow 36a^2-36>0$
$\Rightarrow a\in (-\infty ,-1)\cup (1,\infty )$