Increasing and decreasing functions and the first derivative test We'll use the first derivative to determine where a function is increasing or decreasing, andwhether it has a local maximum or minimum. We are going to take the derivative of f(x) and graph it. Our original function was f(x) = x`2,so our derivative is f′(x) = 2x. Linear goes through the origin, the slope is slightly larger than 1.So what we're going to do islook at the relationships between these two.You will notice in this region right here f isdecreasing. f is decreasing all for x is less than 0. Let's take a look at what's going on over here on this x is less than 0 part. We know the graph of f prime is increasing the entire time. The function f'(x) has a negative value whenever x is less than 0, since a negative derivativecorresponds to a decreasing function. The function f is increasing on an interval when f′ > 0, andthe function is decreasing on that same interval when f′ < 0. Furthermore, f′ = 0 indicates nochange in the values of f. After that, we can also use the sign of the derivative to determine points of inflection, where thegraph changes concavity. We'll call it the graph of g. Let's take a look.And what we want to do istalk about extrema. But this time, we're going to talk about what's called local, also known asrelative extrema. Remember extrema is just the collective word for maxes and mins.This point (the highest point on the graph) is a local maximum. In general, as the graphincreases, it reaches a maximum point- a local maximum. At this point, the graph switches fromincreasing to decreasing, thereby creating a turning point; this is known as a local maximum.And then this part over here-- let's change colors-- that would be a local maximum. It is wherethe graph of f increases rapidly and then decreases slowly. And the local minimum is where thegraph of f decreases rapidly and then increases slowly.So looking at this graph, let's talk about the derivative behavior at these points.
If g increases and then decreases, then g prime will be positive and then negative. This meansthat f has a local maximum when f prime changes from positive to negative and a local minimumwhen f prime changes from negative to positive.So that would change negative to positive. So f as a local min when f prime changes negative topositive.