Section 4 2B: Rolle s Theorem and the Mean Value Theorem The intermediate Value Theorem If f(x) is a continuous function on the closed interval a,b [ ] with end points at ( a,f(a) ) and b,f(b) ( )then 1. If k is a real number value for y betweenf(a) and f(b) then there is at least one value c where a < c < b and and f(c) = k The theorem says that x will take on every the real number value from x = a to x = b and y will take on every real number value from f(a) to f(b). The Mean Value Theorem states there is at least one point where f(c) = k. There may be more than one point where f(c) = k depending on the function and the value of k selected. If f(a) is positive and f(b) is negative (or visa versa) then there are one or more points where f(c) = 0 This guarantees that the function has one or more roots. Lec 4 2A Rolle s Theorem! Page 1 of 6! 2018 Eitel
The Extreme Value Theorem The extreme value theorem states that if f(x) is a real-valued function and f(x) is continuous in the closed interval a,b closed interval a,b [ ] then ff(x) must attain a maximum and a minimum, each at least once on the [ ]. That is, there exist numbers c and d in [a,b] such that: the point ( c,f(c) ) is the maximum point and ( d,f(d) ) is the minimum point on the interval [ a,b]. A continuous function ƒ(x) on the closed interval [a,b] showing the absolute max (red) and the absolute min (blue). Note: Depending on the function there may be several points that are maximum or minimum points. Note: Depending on the function the maximum or minimum points may be at the end points of the closed interval.. The extreme value theorem says that not only is the function bounded, but it also attains its least upper bound at the maximum and its greatest lower bound as its minimum. The extreme value theorem is used to prove Rolle's theorem. Lec 4 2A Rolle s Theorem! Page 2 of 6! 2018 Eitel
Rolle s Theorem The Extreme Value Theorem in the last section states that If f(x) is a continuous function on the closed interval a,b [ ] then f(x) has at least one absolute maximum value and at least one absolute [ ]. These values may occur at an x value on the open minimum value on the closed interval a,b interval (a,b) or at an endpoint of the closed interval. Under certain more specific conditions we can guarantee that an extreme value occurs at an x value in the interior of the interval. Rolle s theorem states the conditions under which an extreme value must occur in the open interval (a,b). Rolle's theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. Rolle s Theorem IF all of the following conditions occur! 1. A function has two different points ( a,f(a) ) and ( b,f(b) ) where f(a) = f(b) or in other words two different points with the same y coordinates.! 2. f(x) is a continuous function on the closed interval [ a,b] or in other words the graph has no breaks or gaps. 3. f(x) is differentiable (the derivative is defined) for all x on the open interval (a,b) or in other words the graph has no sharp points. If all of the above conditions occur then there is at least one real number c in the open interval (a,b) where f (c) = 0 or in other words the graph has a horizontal tangent line at the point c,f(c) (a,b) with a < c < b ( ) in the open interval Lec 4 2A Rolle s Theorem! Page 3 of 6! 2018 Eitel
The first requirement is that f(a) = f(b) The first requirement is that a function has two points ( a,f(a) ) and ( b,f(b) ) where f(a) = f(b)or in other words two different points with the same y coordinates. The two points must both be on a horizontal line. The next requirement is that the function be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). This requires the graph that connects the two points is either a line between the two points or a smooth curve with no breaks or sharp points. If the graph that connects the two points is a line then it must be a horizontal line. Rolle s Theorem is true because if all points in the open interval (a,b) satisfy f (c) = 0 there is at least one that does If the graph that connects the two points increases as it moves to the right from point (a,f(a)) then it must turn downward at some point so that it ends at (b,f(b)). The requirement that the curve is smooth and unbroken forces the curve to have a smooth downward turn at some point between a and b. Rolle s Theorem is true because there is at least one real number c in the open interval (a,b) where f (c) = 0. Lec 4 2A Rolle s Theorem! Page 4 of 6! 2018 Eitel
If the graph that connects the two points decreases as it moves to the right from point (a,f(a)) then it must turn upward at some point so that it ends at (b,f(b)). The requirement that the curve is smooth and unbroken forces the curve to have a smooth upward turn at some point between a and b. Rolle s Theorem is true because there is at least one real number c in the open interval (a,b) where f (c) = 0. Are there other conditions that produce a maximum or minimum point other than the conditions of Rolle s Theorem Rolle s Theorem requires a continuous and differentiable function on the interval (a,b) between two points with the same y coordinates. This creates the conditions that guarantee that a relative maximum or relative minimum point must exist on the interval (a,b) There amy be other other conditions that guarantee that a relative maximum or relative minimum point must exist on the interval (a,b). The two graphs below show a continuous function that is NOT differentiable on the interval (a,b)but do have a relative maximum or relative minimum point. This point occurs at the x value where the derivative is undefined. Clearly conditions other then those required by Roll s Theorem exist the produce maximum or relative minimum points. A complete discussion of how to find maximum or relative minimum points for a function will occur in the next section Lec 4 2A Rolle s Theorem! Page 5 of 6! 2018 Eitel
The Mean Value Theorem (an extension of Rolle s Theorem) If we drop the first requirement of Rolle s Theorem that a function have two different points ( a,f(a) ) and ( b,f(b) ) where f(a) = f(b) we are left with only the 2 conditions that the function be continuous and differentiable. We cannot state that a minimum or maximum point exists form these conditions but we can make a statement about the slope of some point in the interval(a,b).! The Mean Value Theorem If f(x) is a continuous function on the closed interval [ a,b] (the graph has no breaks or gaps) and If f(x) is differentiable (the derivative is defined) for all x on on the open interval (a,b) (the graph has no sharp points) then there is at least one real number c in the open interval (a,b) where f (c) = f(b) f(a) b a or in other words the graph has a point ( c,f(c) ) in the he open interval (a,b) where the slope of the tangent line at x = c equals the slope of the line segment from ( a,f(a) ) to ( b,f(b) ) The slope of the line segment from point ( a,f(a) ) to point ( b,f(b) ) is called the Average (or mean)rate of Change. The slope of the tangent line at the point x =c is called the instantaneous rate of change. It is the rate of change at the instant you are at the point where x = c In terms of related rate and rates of change this theorem says that there is at least one point on the interval (a,b)where the instantaneous rate of change is equal to the average rate of change on the interval [ a,b] The Mean Value Theorem is used to prove many other theorems in Calculus. For this reason it is considered one of the most important theorems in calculus. Lec 4 2A Rolle s Theorem! Page 6 of 6! 2018 Eitel