Our knowledge of differentiation can be used to determine how a graph looks like without actually plotting it's points. For that, you need to understand the concept of Critical Points and Maximum & Minimum.
By the end of this lesson, you will be able to apply the first derivative test on an equation to determine if it has a local maxima or minima.
Estimated time: 20 minutes
From the video, you learn that Critical Point is a point where the derivative of a function f(x) is equal to zero, that is f'(x) = 0. The slope of the tangent line at the critical point is horizontal. The left graph below has 1 critical point and the right graph has 2 critical points.
Now that you understand what is a critical point, you will be able figure out how a graph look like by using the first derivative test.
From the video, you learnt that the using the first derivative test, you can determine whether a graph has a local maxima or minima .
For the left graph below, the sign of f'(x) changes from + to − as x increases through the point x = xo, then f(x) has a local maximum at x = xo.
For right graph below, the sign of f'(x) changes from − to + as x increases through the point x = xo, then f(x) has a local minimum at x = xo.
Worked example #1 using first derivative test to find local maximum and minimum of a function