00:01
For this problem, we are asked to determine if the mean value theorem will apply to the function f of x equals x power of 2 over 3 over the closed interval from 0 to 1.
00:09
So our, and if it does, we are then asked to find c such that f prime of c equals f of b minus f of a over b minus a over b minus a.
00:16
So to begin, we need to check, is f of is f continuous on our interval? well, x power of 2 over 3 is continuous everywhere.
00:25
So that is satisfied.
00:27
Then we need to determine if it has a continuous derivative.
00:30
Well, the derivative is going to be 2 over 3 times x to the power of 1 over 3.
00:37
Now, we'll note that that is not continuous, or it's not defined at, oops, i did not mean to switch to a red pen, it's not defined at x equals 0, but we need f to only be differentiable on the open interval from a to b.
00:53
This is differentiable for everywhere between 0 and 1, but not at the left end point, but that's not a problem...
