Page 92 - Calc for the Clueless

So our approximation is

Mean Value Theorem

The mean value theorem (MVT) is a very important theorem in differential calculus that periodically pops up to
give us more useful information.

The mean value theorem: Let f(x) be continuous on [(a,b)]. Let f'(x) exist on (a,b). There exists a point c
between a and b such that f'(c)= f(b) - f(a)/b - a.

Let us translate. There are no breaks between a and b, including both ends. The derivative exists except at
possibly the left and right ends. Otherwise it is smooth, f'(c) is the slope at c. What the heck is the rest? If x = a,
the y value is f(a). If x = b, the y value is f(b). The two end points are (a,f(a)) and (b,f(b)). The slope of the line
joining the two end points is [f(b) -f(a)]/(b - a).

The theorem says there is at least one point between a and b where the curve has the same slope as the line
joining the end points of the curve.

Question 1

Can there be more than one such point? The answer is yes, but one and only one is guaranteed by the theorem.

Question 2

Can there be such a point if the continuity or differentiability condition does not hold? The answer is yes, but
there also may not be such a point. In both the cases illustrated below, there is no point on the curve where the
slope is the same as the slope of the line joining the end points of the curve.