2.
Let $$f$$ be the function defined by $$f(x)=x^3+x-4$$. What is the value of $$c$$ for which the instantaneous rate of change of $$f$$ at $$x=c$$ is the same as the average rate of change of $$f$$ over $$[-1,3]$$?
$$$
\begin{aligned}
f^\prime(x)&=3x^2+1\\
\text{Ave slope }&=\frac{f(3)-f(-1)}{3-(-1)} = \frac{26-(-6)}{4}=\frac{32}{4}=8\\\\
\text{We need } f^\prime(x)&=3x^2+1=8\\
x&=\sqrt{\frac{7}{3}}
\end{aligned}
$$$