# Analyzing functions with calculus

Contents

Let's put all of our differentiation abilities to use, by analyzing the graphs of various functions. As you will see, the derivative and the second derivative of a function can tell us a lot about the function's graph.

Critical points are points where a function may obtain their minimum or maximum value. They play a critical role (pun intended) in analyzing the increasing and decreasing intervals of functions, and in finding their minimum and maximum points.

Analyze functions to find the intervals in which they are increasing or decreasing.

Analyze functions to find their relative extrema (i.e. their relative minimum and maximum points).

Analyze functions to find their absolute extrema (i.e. their absolute minimum and maximum points).

Review your understanding of increasing and decreasing intervals, and extremum points, with some challenge problems.

Concavity describes the shape of a graph as it increases or decreases: a graph that's concave up is shaped like a cup, U, and a graph that's concave down is shaped like a cap, ∩. Learn more about concavity and how it relates to a function's second derivative.

Points of inflection are points where the function changes concavity. Learn how to find and analyze them.

Combine the tools provided by differential calculus with other algebraic tools in order to obtain detailed sketches of various functions.

Review your understanding of concavity and points of inflections with some challenge problems.