Blog 4

Hello everyone,
my name is professor Maragkos and in today's class i'll be teaching what the inflection points are and how to find them.
First of all, we have to review what we said last class about critical points. Critical point is a point p in the domain of f where f'(p) = 0 or f'(p) is undefined, or in other words, a point on the graph of a function where the slope changes concavity . Below, i have included a few examples of critical points.

In figure 4.3, we can see that there is only one critical point, in figure 4.4, there are no critical points, and finally in figure 4.5 there are 4 critical points marked.



This brings us to the main topic of today's lecture, which is the inflection points of a graph. Inflection point is a point at which the graph of a function f changes concavity. In other words, we can locate an inflection point by looking at the second derivative of that function. An inflection point will be where the f'' is zero or undefined, or in other words the sign of f'' will be positive on one side of the inflection point and negative on the other, as shown in the graph below.

Let's see an example so you can understand this concept better.
ex. Find the inflection points of f(x) = x^3-9x^2-48x+52

Step 1
Calculate first and second derivatives
f '(x)= 3x^2-18x-48
f ''(x)= 6x - 18
so f ''(x)=0 when x=3

Step 2
Graph the function and determine the inflection points

As we can see from the graph of f(x), concavity changes at x=3, so x=3 is an inflection.
Please, if you have any questions, now is the time to ask.