I am having problems understanding how to find the maximum value from a rate of change (derivative) function. The rate of change of Volume with respect to time is $\frac{dv}{dt}=1000- 30t^2 +2t^3$, $0 \le t \le 15$
How do I find the maximum rate of change? the answers is where $t=0$ and $t=15$, but I can't see how this is done?
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$\begingroup$Relative extrema occur at endpoints or critical points. Critical points occur where a function's derivative is $0$ or undefined.
The maximum of $\frac{dv}{dt}$ is where $\frac{d^2v}{dt^2}=0$ or is undefined.
\begin{align*} \frac{d^2v}{dt^2} &= -60t + 6t^2 =0 \\ 0 &= 6t(t - 10) \\ t&=0, \ t=10 \end{align*}
So $\frac{dv}{dt}$ has critical points at $t=0,10$. Now you have to find $\frac{dv}{dt}(0)$, $\frac{dv}{dt}(10)$, and $\frac{dv}{dt}(15)$ and see which one has the highest value to find the max.
\begin{align*} \frac{dv}{dt}(0)\ \ &= 1000 \\ \frac{dv}{dt}(10) &= 0 \\ \frac{dv}{dt}(15) &= 1000 \end{align*}
On the domain $[0,15]$, $\frac{dv}{dt}$ has a maximum value of $1000$ at $t=0,15$.
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