00:01
In this question, we're asked to find all of the possible values of c that satisfy the mean value theorem for a certain function over a certain interval.
00:10
I've written down the mean value theorem in green here, which tells us that if some function is continuous on a closed interval and differentiable on an open interval, then there is some number c that is equal to the slope of the tangent line.
00:26
In other words, there's going to be a point where the slope of a tangent line is going to be equal to the slope of a secant line.
00:34
So, looking at our function, because our function is a polynomial, we know that it must be continuous, and we know that it will be differentiable on this interval here from 0 to 2.
00:47
Therefore, we want to find, therefore we want to see, we want to find the values of c, such that we have f of 2, so we have the slope of the secret line at whatever this point is equal to that point, equal to that.
01:06
So we can plug into this function here.
01:08
So i have 2 times 8, which is 16 plus 4, 7, that's 27, minus 7 over 2, which will be equal to 20 over 2 or 10...