Calculus: things calculus can do: local extrema

Tutoring calculus, you might hear curiosity about what it’s used for. The tutor mentions local extrema.

In high school algebra, one can learn to expect that the graph y=2x^3 +10x^2 -11x -14

might have one relative minimum and one relative maximum. Yet, where do they occur?

Local high points and local low points are also known collectively as relative extrema. A relative maximum, for instance, means that the graph might take on a higher value somewhere else, but immediately either side of this one, it’s lower.

To find the relative extrema of a function, one solves for when its derivative is zero. (For background about the idea of the derivative, see my post from yesterday.) The concept is that at a local maximum, since the value on either side needs to be lower, the direction of the curve must be horizontal, neither climbing nor falling, at that exact point. (Note that “value” means y-value.) The same concept extends to a local minimum: there, too, the derivative will be zero. This assumes the derivative exists, of course, but for a polynomial it will.

The derivative of y=2x^3 +10x^2 -11x -14 will be a quadratic, which will have two zeros: one imagines one of them will be for a local minimum, the other, at a local maximum.

In coming posts I plan to continue about how to arrive at said derivative, then find its zeros, etc.

Source:

Larson, R.E., Hostetler, R.P. (1989). Calculus part one, third edition. D.C. Heath and Company.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.