Question

If the function $f$ be given by $f(x)=3 x^4+4 x^3-12 x^2+12$, then

• A. local maximum value of $f$ is 12
• B. local minimum values of $f$ are 7 and $-20$
• C. Both(a) and (b) are correct
• D. Both(a) and (b) are incorrect

Answer 1

Answer: (C)

We have,
$f(x)=3 x^4+4 x^3-12 x^2+12$
or $ f^{\prime}(x) =12 x^3+12 x^2-24 x=12 x(x-1)(x+2) $
$ f^{\prime}(x) =0 $ at $ x=0, x=1 $ and $x=-2$
Now, $f^{\prime \prime}(x)=36 x^2+24 x-24=12\left(3 x^2+2 x-1\right)$
or $\begin{cases}f^{\prime \prime}(0)=-12 < 0 \\ f^{\prime \prime}(1)=48>0 \\ f^{\prime \prime}(-2)=84>0\end{cases}$
Therefore, by second derivative test, $x=0$ is a point of local maxima and local maximum value of $f$ at $x=0$ is $f(0)=12$, while $x=1$ and $x=-2$ are the points of local minima and local minimum values of $f$ at $x=-1$ and $-2$ are $f(1)=7$ and $f(-2)=-20$, respectively.