00:01
Suppose f of x equals 6 times the square root of x minus 6 on the interval 16 to 36.
00:06
We want to find the value of c in the open interval 16 to 36 guaranteed by the mean value theorem.
00:14
Now what is this mean value theorem? mean value theorem states that if a function is continuous on the closed interval a, b and differentiable in the open interval a, b, then there will be a number c in the open interval a, b.
00:31
Such that f prime of c equals the average rate of change in the interval a, b, or that's f of b minus f of a over b minus a.
00:43
So since we're looking for the value of c in the open interval 16 to 36, then we're going to use this part of the theorem.
00:52
First, you need to find the derivative of the function.
00:55
Now f prime of x equals 6 times square root of x is x to the 1 half so that's 1 half times x raised to negative 1 half and then that's the same as 3x to the negative 1 half or that's 3 over square root of x.
01:21
So when x is equal to c.
01:23
Our f prime of c equals 3 over the square root of c...