00:01
In this problem, we are given the following function.
00:07
F of x equal to x minus 3 squared on the interval from 2 to 5.
00:15
And we have three questions to answer.
00:18
So in the first one, we are going to find the average value of this function on this given interval.
00:25
So let's call this, or let's denote the average by this angle bracket notation.
00:30
So we have 1 over 5 minus 2 integral from 2 to 5 dx f of x.
00:39
This is the general formula for the average for this function.
00:44
So we have 1 over 3 integral from 2 to 5 dx x minus 3 squared.
00:53
So 1 over 3 x minus 3 cube over 3 evaluated between 2 and 5.
01:01
And we find 1.
01:03
Next, we are going to find some number c such that the average of this function is equal to this function evaluated at this number c.
01:21
So we have 1 equal to c minus 3 squared.
01:28
So we obtain c equal to 2 or 4.
01:32
So we should be careful with the domain of this function.
01:36
The domain is all the real numbers from 2 to 5 with 2 and 5 included.
01:43
So these solutions are both on this interval...