Question

If the function $f$ be given by $f(x)=3x^4+4x^3-12x^2+12$, 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,
$f(x)=3x^4+4x^3-12x^2+12$
or $f^{\prime }(x)=12x^3+12x^2-24x=12x(x-1)(x+2)$
$f^{\prime }(x)=0$ at $x=0,x=1$ and $x=-2$
Now, $f^{\prime \prime }(x)=36x^2+24x-24=12\left(3x^2+2x-1\right)$
or $\{\begin{array}{l}{f}^{\prime \prime }(0)=-12<0\\ f^{\prime \prime }(1)=48>0\\ f^{\prime \prime }(-2)=84>0\end{array}$
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.