For the period from 2000 to 2008, the percentage of households in a certain country with at least one DVD player has been modeled by the function
$f(t) = \frac{87.5}{1 + 17.1e^{−0.91t}}$
where the time $t$ is measured in years since midyear 2000, so $0 \leq t \leq 8$. Use a graph to estimate the time at which the number of households with DVD players was increasing most rapidly. Then use derivatives to give a more accurate estimate. (Round your answer to two decimal places.) $t = ?$
On
Okay, so you have $$\begin{align} f'(t)&=87.5\cdot\frac{17.1\cdot -.91e^{-.91t}}{1+17.1e^{-.91t}} \\ &=87.5\cdot\frac{17.1\cdot -.91e^{-.91t}-.91+.91}{1+17.1e^{-.91t}} \\ &=87.5\cdot.91\left(\frac{1}{1+17.1e^{-.91t}}-1\right) \\ &=87.5\cdot.91(f(t)-1) \end{align}$$
That should make it easier to solve for $f''(t)$, and hence $f''(t)=0$. You can take it from here.
Because you are asked to use a graph you should start by plotting $f(t)$ in mathematica, matlab, maple etc. If you don't have access to any of this software take a look at wolframalpha. Then, the time it is increasing most rapidly should be where the slope is greatest. You are right, you will need to find an inflection point which is the deravitave of the slope, or the second derivative of the function and find the critical values, i.e. set it equal to zero and solve for $t$. And you are right, it might get ugly, but it will reduce.