CRITICAL POINTS, EXTREME VALUE THEOREM AND MEAN VALUE THEOREM

EXTREMA A huge topic in Calculus is finding relative (or local) extrema. Check out this graph: The tops of the mountains are relative (local) maximums because they are the highest points in their little neighborhoods (relative to the points right around them):

Suppose you're in a roomful of people (like your classroom.) Find the tallest person there. (It's usually a guy.) He is the relative (local) max of that room. Specifically, he's the tallest, relative to the people around him. But, what if you took that guy to an NBA (National Basketball Association) convention? There would be a lot of guys who beat him.

Look back at the graph... Relative extrema (maxs & mins) are sometimes called local extrema. Other than just pointing these things out on the graph, we have a very specific way to write them out. Officially, for this graph, we'd say: f has a relative max of 2 at x = -3. f has a relative max of 1 at x = 2. The max is, actually, the height... the x is where the max occurs. So, saying that the max is (-3, 2) would be unclear and not really correct.

Now, for the relative minimums... These are the bottoms of the valleys: Relative mins are the lowest points in their little neighborhoods. f has a relative min of -3 at x = -1. f has a relative min of -1 at x = 4.

So, how many relative mins and maxs does the typical polynomial have? Don't know? When in doubt, draw pictures! Let's draw some possible shapes of Remember, we use how many times the function can cross (the degree) to guide us. Hmm... It looks like an polynomial can have, at most, 3 relative extrema. A polynomial of degree n can have, at most, n - 1 relative extrema.

CONTINUITY So far we have been discussing functions that are differentiable. What about non- differentiable functions? How do we know when a function is not differentiable? A function fails to be differentiable at a point if: The point is a sharp corner point. In this case, the left-hand derivative and the right-hand derivative are different and therefore the limit of the difference quotient does not exist. The tangent line is vertical at the point since vertical lines have no slopes. The function is discontinuous at a point (hole).

Theorem If a function f(x) is differentiable at x = c, then it is continuous there. A continuous function need not be differentiable. That is, the converse of the above theorem is not true in general, so be careful not to consider all continuous functions to be differentiable.

CRITICAL POINTS One of the important applications of the derivative is finding critical points. A critical point is a location (x=c) in a function where the function: f(c) exists f ’(c) = 0 {Has a derivative equal to zero (slope of tangent is horizontal)} f ’ (c) DNE {Has a derivative that does not exist (slope of tangent is vertical)}

INSTRUCTIONS FOR FINDING THE CRITICAL POINTS: 1.Take the derivative of the function. 2.Factor the derivative. You may not be able to factor the derivative but if you can factor it, this will make finding the critical points easier 3.Set the derivative equal to zero 4.Solve for x 5.Check that f(c) exists (look at domain restrictions!)

EXTREME VALUE THEOREM An important application of critical points is in determining possible maximum and minimum values of a function on certain intervals. The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. It states the following: If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b].

The procedure for applying the Extreme Value Theorem is to: 1.establish that the function is continuous on the closed interval 2.determine all critical points in the given interval 3.evaluate the function at these critical points and at the endpoints of the interval. The largest function value from the previous step is the maximum value, and the smallest function value is the minimum value of the function on the given interval.

ROLLE’S THEOREM Rolle’s Theorem Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where f ‘ (c) = 0. This means that Rolle’s theorem claims the existence of a point at which the tangent to the graph is parallel to the secant, provided the secant is horizontal.

MEAN VALUE THEOREM (AVERAGE VALUE THEOREM) The Mean Value Theorem is one of the most important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that In the graph, the tangent line at x = c [f ‘ (c)] is equal to the slope of [a, b]; the secant line.

This theorem is beneficial for finding the average of change over a given interval. For instance, if a person runs 6 miles in an hour, their average speed is 6 miles per hour. This means that they could have kept that speed the whole time, or they could have slowed down and then sped up (or vice versa) to get that average speed. This theorem tells us that the person was running at 6 miles per hour at least once during the run. If we want to find the value of c, we 1.Find (a,f(a)) and (b,f(b)) 2.Use the Mean Value Theorem 3.Find f'(c) of the original function 4.Set it equal to the Mean Value Theorem and solve for c.

HOMEWORK p. 169 (3-9odd, 17, 19, 25, 27, 35, 41, 43, 54, 57-60) p. 176 (13, 15, 21, 29, 37-47odd, 75) Outline 3.3 and 3.4