The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation $D ( t ) = - 10.82 t ^ { 3 } + 209.508 t ^ { 2 } - 167.402 t + 9774.134$ donors, where t is the number of years after 1975. Find the inflection point(s) from $t = 0$ through $t = 17$ , if any exist.
A)There are no inflection points from $t = 0$ through $t = 17$ .
B)There is one inflection point at $t = 6.45$ .
C)There are inflection points at $t = 0$ and $t = 17$ .
D)There is one inflection point at $t = 0.15$ .
E)There are inflection points at $t = 0$ , $t = 0.15$ , and $t = 17$ .

Question

Quizplus · Accepted Answer

To find the inflection points, we need to find where the second derivative of the given function changes sign. The second derivative of $D(t) = -10.82t^3 + 209.508t^2 - 167.402t + 9774.134$ is $D''(t) = -64.92t + 419.016$. Setting this equal to zero to find critical points gives $t = 419.016 / 64.92 \approx 6.45$. This is the only critical point for the second derivative in the given interval, indicating a single inflection point at $t = 6.45$.

The Number of People Who Donated to a Certain Organization D(t)=−10.82t3+209.508t2−167.402t+9774.134D ( t ) = - 10.82 t ^ { 3 } + 209.508 t ^ { 2 } - 167.402 t + 9774.134D(t)=−10.82t3+209.508t2−167.402t+9774.134

The Number of People Who Donated to a Certain Organization $D ( t ) = - 10.82 t ^ { 3 } + 209.508 t ^ { 2 } - 167.402 t + 9774.134$