Study Guide

Second Derivatives and Beyond - Special Points

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Special Points

Special points are those random points awarded to the guy who invited us over to play board games. Blast you, house rules. They never seem to work in our favor.

In derivative land, if we have a function f whose first two derivatives exist (at least most of the time), there are some special types of points that we need to be able to find. Remember that the roots of a function are those x-values where the function value is equal to zero.

Be Careful: We use the word "point" in these sections to refer to an x-coordinate all by itself. When it comes time to graph things, we will need to find both the x and y coordinates in order to have a point that we can graph.

  • Critical Points

    We say x = c is a critical point of the function f if f (c) exists and '(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interesting. When climbing Mount Everest, we might say that we've reached the critical point when we've reached the summit. It's where we can enjoy the view while feasting on granola.

    Be Careful: A critical point of a function f is a value in the domain of f at which the derivative is 0 or undefined.

    It's also possible for a function to have no critical points at all.

    To find the critical points of a function f we

    • take the derivative of f
    • find any places where ' is undefined but f is defined, and finally
    • find the roots of the derivative.

    We can often check our answers by graphing the function and making sure it looks like it has critical points in the right places. We know what a graph looks like at a spot where the derivative doesn't exist:

    and we know what a graph looks like when it has a derivative of 0 (horizontal tangent line):

  • Points of Inflection

    A point of inflection or inflection point, abbreviated IP, is an x-value at which the concavity of the function changes. In other words, an IP is an x-value where the sign of the second derivative changes. It might also be how we'd describe Peter Brady's voice.

    The usual way to look for inflection points of f is to

    • find "
    • find all x-values where " is zero or undefined, and
    • check each such x-value to see if the sign of " changes there.

    Again, we can use graphs to check our work. An inflection point where the function goes from concave up to concave down looks something like this:

    An inflection point where the function goes from concave down to concave up looks something like this:

    While any point at which ' is zero or undefined is a critical point, a point at which " is zero or undefined is not necessarily an inflection point. You can think of the points where " is zero or undefined as possible inflection points, but then you need to check each possible inflection point to see if it's a real inflection point.

    Be Careful: Just because "(c) = 0 or is undefined doesn't mean c is an inflection point. " must have different signs on either side of c.

    There are two main ways to figure out what the sign of " is doing on either side of a possible inflection point c.

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