Calculus Tutorial

Intro

In Single-Variable Calculus, one of the core concepts that is always tested is determining extrema, inflection points, and saddle points. This is ESPECIALLY the case on the AP Calculus AB/BC exam. This article is a review of the process of finding extrema/inflection points given a function f(x), and understanding the graphical representations of these points.

Sample Problem

How do we determine extrema and inflection points, understand their representation on graphs, and write answers that the AP graders approve of?

Solution

Relative/Absolute Maximum

1. Take the derivative f'(x)
2. Set the derivative equal to zero and find the x-values that solve the equation(these are critical points – possible extrema points).
3. Evaluate f'(x) at x-values less and greater than the x-value of the critical point.
4. If f'(x) changes from positive to negative, there is a relative maximum at that critical point. If f'(x) changes from negative to positive, there is a relative minimum at that critical point. (NOTE: this is the exact wording AP graders want!).
5. If an interval is given, you can also test for absolute extrema. If f(x) for these x-values is greater than that of the relative maxima/minima, they are absolute maxima/minima. Or else, the relative maxima/minima are also the absolute maxima/minima.

Inflection Points

1. Find f”(x)(the second derivative of f(x))
2. Set f”(x) = 0. Find the x-values that solve this equation(possible inflection points).
3. Evaluate f”(x) at x-values less and greater than that of the possible inflection point.
4. If f”(x) changes from positive to negative OR negative to positive, f(x) has an inflection point at x(again, AP justification).

Graphical Representation!

If you are given a graph of f'(x)(very likely to appear on the AP Exam!) –

When f'(x) = 0, those x-values represent points (x, f(x)) that are possible maxima/minima. If f'(x) changes from positive to negative, there is a relative maximum at that critical point. If f'(x) changes from negative to positive, there is a relative minimum at that critical point.

When points on f'(x) are maxima/minima, they are likely inflection points. At these points f”(x) = 0. Check if f'(x) is increasing or decreasing at x-values less and greater than that of the possible inflection point(Draw tangent lines to the curve). If f'(x) increases than decreases or decreases than increases, there is an inflection point at that value x.

If you are given a graph of f”(x) –

When f”(x) = 0, those x-values represent points (x, f(x)) that are possible inflection points. If f”(x) changes from positive to negative, there is an inflection point. If f'(x) changes from negative to positive, there is an inflection point.

Practice!
1. Find the relative/absolute maxima and minima as well as the inflection points of f(x) = x^3 – 8x^2 + 16x – 2

2. Given the f'(x) and f”(x) graphs, label maxima/minima/inflection points 