The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation

D(t)=-10.61t^(3)+208.808t^(2)-168.202t+9775.234 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=6.56.

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.

I thought it was B.

An inflection point is a point on a curve at which the second derivative changes sign. You must find point where second derivative = 0

Go on:

wolframalpha dot com

When page be open in rectangle type:

second derivative of -10.61t^(3)+208.808t^(2)-168.202t+9775.234

and click option =

When you see result click option:

Show steps

Then in rectangle type:

solve 417.616-63.66t=0

Inflection point is:

t=6.5601

Thanks, i thought it was B. your awesome.

To find the inflection point(s) of a function, we need to determine the points where the concavity changes. In other words, we are looking for values of t for which the sign of the second derivative changes.

To find the second derivative of the given function, D(t), we need to differentiate it twice. Let's start by finding the first derivative:

D'(t) = -31.83t^2 + 417.616t - 168.202

Now, let's find the second derivative by differentiating D'(t) with respect to t:

D''(t) = -63.66t + 417.616

Now, we can set D''(t) equal to zero and solve for t to find the potential inflection point(s):

-63.66t + 417.616 = 0

Let's solve this equation:

-63.66t = -417.616

t = -417.616 / -63.66

t ≈ 6.56

So, the potential inflection point is at t ≈ 6.56.

However, we need to check whether this point falls within the interval from t = 0 to t = 17. Since 6.56 is between 0 and 17, we can conclude that there is indeed one inflection point at t = 6.56.

Therefore, the correct answer is b. There is one inflection point at t = 6.56.