\(f(x)\) is a cubic polynomial and the curve represented by \(y=f(x)\) passes through the origin. The function \(y=f(x)\) is an increasing function in
\[
(-\infty,0)\cup(1,\infty)
\]
and a decreasing function in
\[
(0,1).
\]
If
\[
f'(2)=6,
\]
then the \(c\) of Lagrange's Mean Value Theorem on the interval \([0,2]\) satisfies the equation
Show Hint
If a cubic function is increasing and decreasing on specified intervals, first determine the zeros of
\[
\boxed{f'(x)}
\]
from the sign changes. Then apply Lagrange's Mean Value Theorem using
\[
\boxed{f'(c)=\dfrac{f(b)-f(a)}{b-a}.}
\]