Question

Let $f\left(x\right)=x^3+3x^2-9x+2$.Then

• A. $f\left(x\right)$ has a maximum at $x=1$
• B. $f\left(x\right)$ has neither a minimum nor a maximum at $x=-3$
• C. $f\left(x\right)$ has a minimum at $x=1$
• D. none of these

Answer 1

Answer: (C)

$f\left(x\right)$ has a minimum at $x=1$

$f\left(x\right)=x^3+3x^2-9x+2$
$f^{\prime }\left(x\right)=3(x^2+2x-3)=3(x+3)(x-1)=0$
$\Rightarrow x=-3,1$
$f^{\prime \prime }\left(x\right)=6(x+1)$
now $f^{\prime \prime }(-3)<0$
and $f^{\prime \prime }\left(1\right)>0$
Therefore $f\left(x\right)$ is maximum at $x=-3$ and minimum at $x=1$
Ans: C