Given a graph over a domain, identify relative and absolute extrema
Compute the Critical Points of a function
Determine the concavity of a function at a point
Apply the 2nd derivative test to determine whether a critical point is a relative max or min
Video3.1.1.Absolute vs Relative Extrema.
In this video we come up with definitions for the notions of absolute and relative maximums and minimums and visualize this with graphs of functions.
Post-video reflection.
AWhat kinds of polynomials have an absolute max or min on the real numbers? What about on a closed interval like [0,1]? We’ll answer that latter question more clearly during class.
Video3.1.2.Finding Relative Extrema.
Now we turn to the question of how to find a maximum or minimum. We will define a concept called the critical point which provides a necessary condition to be a max or min.
Post-video reflection.
If f(x) is not defined at x = c, could c be a critical number?
Video3.1.3.Concavity and the 2nd Derivative Test.
The concavity of a function can be determined using the second derivative test. If the second derivative is positive at a critical point, the function is concave up and the critical point is a relative minimum. If the second derivative is negative, the function is concave down and the critical point is a relative maximum.We have previously seen how the first derivative told us intervals where a function was increasing and decreasing. Now we look to what the second derivative can tell us, a property we refer to as concavity.
Post-video reflection.
Both the 1st and 2nd derivative provide us information about whether a critical point is a max or min. Which method do you prefer? Do you always prefer it, or does it depend?
