Given the function #f(x)=((x)/(x+4)) #, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,8]?
The Mean Value Theorem states that if
The given function, as mentioned above, is continuous and differentiable everywhere except at
The truth of the Mean Value Theorem thus implies that the conclusion of the Mean Value Theorem will be satisfied for this function on this interval. That is, there will exist a number
It's not necessary to do, but we can also attempt to find the value(s) of
The value(s) of
Only one of these is in the interval
The mean value theorem states that, if f(x) is continuous on an interval [a,b] and differentiable on that same interval, then there is at least one point on that interval (call it c) such that
Basically, if you connect points (a,f(a)) and (b,f(b)) with a line, then the slope of that line will be the same as the derivative of f(x) at some point in that interval.
so next we'll find f'(x). Using the rule for the derivative of the quotient of 2 functions, we have
Set this equal to
And now, all we have to do is solve this quadratic equation. It turns out to have roots -11.75 and 3.75.
The first one is NOT in interval [1,8] but the second one is. Therefore, this function satisfies the mean value theorem.