Determining Concavity

#5.6 Determining Concavity of Functions over Their Domains

At this point, you should know from the previous sections that the first derivative of a function can tell you a lot of information about the function, such as where it increases or decreases and where there is a minimum or maximum. Well, can the second derivative of a function tell us any information about the function? Yes, it can! Let’s learn about how we can determine the concavity of a function using its second derivative. ⬇️

#🧗‍♀️ Determining Concavity

First off, what is concavity? In calculus, a function is said to be concave up if it faces upward and concave down if it faces downward. More technically speaking…

• If the slopes of the lines tangent to the function are increasing or the function’s derivative is increasing, then the function is concave up.
• If the slopes of the lines tangent to the function or the function’s derivative is decreasing, then the function is concave down.

Examples of concave up and concave down

Image Courtesy of The Organic Chemistry Tutor

#🥈 The Second Derivative

Based on this definition of concavity, we can start to see how concavity can be de...