Question

Let $f\left(x\right)=x^3-3x^2-9x+9,$ then which of the following(s) is(are) correct

• A. $f\left(x\right)$ has a local minima at $x=3$
• B. $f\left(x\right)$ has a local maxima at $x=-1$
• C. $f\left(x\right)$ has a local minima at $x=-1$
• D. $f\left(x\right)$ has a local maxima at $x=3$

Answer 1

Answer: (A), (B)

$f\left(x\right)$ has a local maxima at $x=-1$

Given : $f\left(x\right)=x^3-3x^2-9x+9$
For critical point : $f^{\prime }\left(x\right)=0$
$\Rightarrow 3x^2-6x-9=0$
$\Rightarrow x^2-2x-3=0$
$\Rightarrow x=-1,3$
Now, $f^{\prime \prime }\left(x\right)=6x-6$ $f^{\prime \prime }(-1)=-12<0$
So local maxima at $x=-1$
$f^{\prime \prime }\left(3\right)=12>0$
So local minima at $x=3$