Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero. All relative maxima and relative minima are critical points, but the reverse is not true.
I want to talk about a really important concept in Calculus called the critical point here's the definition, let f be a function and let c be a point in this domain. We call c a critical point if f prime of c is 0 or if f prime of c is undefined. We want to look for critical points because it'll be really important when we started graphing functions using their derivatives but let's look at an example where we find some critical points. Consider the function g of x equals 3x to the fourth minus 20x cubed plus 17 and I have that function graphed here. I'm asked to find the critical points and explain their Geometric significance so first I'm going set out to find the critical point.
And you'll notice the critical point involves the derivative so the first thing I want to do to this function is take its derivative. So it's a polynomial really easy to differentiate the derivative of 3x to the fourth is 12x cubed, the derivative of 20x cubed is 60x squared and the derivative of 17 is 0. So this is my derivative nice and easy let me factor this, it'll always be easier to find critical points if I factor the derivative and so I'm going to pull out the common factor of 12 and x squared 12x squared and that leaves an x and a 5. And so this is my derivative factor now remember I want to look for critical points and critical points are where the derivative equals 0 or where it's undefined. Now this derivative is never undefined so I need to find where it's 0, so I set it equal to 0 and solve for x and this is really easy because it's already factored and that's why I factored it. So that tells me that x=0 or x=5 these are the 2 critical points.
Now let me show you what the critical points look like on the graph. So just going back here I have the graph I didn't mark any of these points but since you remember that the derivative gives us the slope of a tangent line and the critical points are points where the slope of the tangent line is 0, these are the critical points here x=0 and x=5. And their Geometric significance is that these are the points where the derivates or where the tangent is horizontal. So if you were to draw a tangent line at these 2 points you get a horizontal line. So once again the critical point is a point where the derivative equals 0 or is undefined and it's significance at least in this example is that these are the points where the tangent is horizontal.