Question

If the slope of the curve $y=-x^3+3x^2+9x-27$ is maximum at a point $P(\alpha ,\beta )$ on it, then which of the following is/are true

• A. ${\alpha }^2+{\beta }^2=255$
• B. ${\alpha }^2+{\beta }^2=257$
• C. Maximum slope is $12$
• D. Maximum slope is $-16$

Answer 1

Answer: (B), (C)

Maximum slope is $12$

$y=-x^3+3x^2+9x-27$
Slope, $m=f\left(x\right)=-3x^2+6x+9$
$\Rightarrow f^{\prime }\left(x\right)=-6x+6$ and $f^{\prime \prime }\left(x\right)=-6<0$
$\therefore m$ is maximum when $f^{\prime }\left(x\right)=0$
$\Rightarrow x=1$
$\Rightarrow y=-16$
$\therefore P(\alpha ,\beta )\equiv (1,-16)$
$\Rightarrow {\alpha }^2+{\beta }^2=257$
$\Rightarrow$ maximum slope $=f\left(1\right)=12$