QuestionFind all numbers $x$ in $(-1,2)$ where $f'(x)=3x^2+4x-1$ equals the average rate of change of $f(x)=x^3+2x^2-x+6$.

Studdy Solution
We have two possible solutions for $x$, but we need to check if they lie in the open interval $(-,2)$.The solutions are $x=\frac{-2+\sqrt{19}}{3}$ and $x=\frac{-2-\sqrt{19}}{3}$.The numerical values of these solutions are approximately $x=.897$ and $x=-2.230$, respectively.
Only $x=\frac{-2+\sqrt{19}}{3}$ lies in the open interval $(-,2)$.
So, the only value of $x$ in the interval $(-,2)$ at which the derivative equals the average rate of change of $f$ on the interval is $x=\frac{-2+\sqrt{19}}{3}$.