Question:
Let $f(x)=x^{3}+3 x^{2}-9 x+2$. Then, $f(x)$ has
(a) a maximum at $x=1$
(b) a minimum at $x=1$
(c) neither a maximum nor a minimum at $x=-3$
(d) none of these
Solution:
(b) a minimum at $x=1$
Given: $f(x)=x^{3}+3 x^{2}-9 x+2$
$\Rightarrow f^{\prime}(x)=3 x^{2}+6 x-9$
For a local maxima or a local minima, we must have
$f^{\prime}(x)=0$
$\Rightarrow 3 x^{2}+6 x-9=0$
$\Rightarrow x^{2}+2 x-3=0$
$\Rightarrow(x+3)(x-1)=0$
$\Rightarrow x=-3,1$
Now,
$f^{\prime \prime}(x)=6 x+6$
$\Rightarrow f^{\prime \prime}(1)=6+6=12>0$
So, $x=1$ is a local minima.
Also,
$f^{\prime \prime}(-3)=-18+6=-12<0$
So, $x=-3$ is a local maxima.