Suppose you are given a formula for a function $ f $.
(a) How do you determine where $ f $ is increasing or decreasing?
(b) How do you determine where the graph of $ f $ is concave upward or concave downward?
(c) How do you localte inflection points?
(a) Use the Increasing/Decreasing (I/D) Test.
(b) Use the Concavity Test.
(c) At any value of $x$ where the concavity changes, we have an inflection point at $(x, f(x))$
the graph of F A bex to determine when the graph is increasing. You can see that from this point here to this point here, the graph of F of X is increasing F of X is increasing when it's first derivative is cratered and zero. So F is increasing when it's first derivative is positive, F is decreasing. So our function F will be decreasing when it's derivative, F. Prime of X Is negative or less than zero. So function will be increasing when it's derivative is positive and the function will be decreasing when it's derivative is negative. If you want to uh determine where the graph is concave upward. Concave upward looks like this kind of like a you opening upwards. So this is a portion of the graph that is concave upwards in order to determine where the function would be concave upwards. We need the second derivative to be positive. And if you want to determine where the graph is concave downwards, concave downwards, concave downwards would be this portion of the graph. Whereas concave upwards kind of starts around here and goes to about maybe at this point right here, we'll talk about this red point is a very special point in a minute. Uh But this portion of the graph is concave upwards. The second derivative for these X values will be positive. This portion of the graph is concave downwards. Okay, kind of like an upside down U. It's opened up facing down. So this is concave downwards for X values. Uh in this region where the graph is concave down, the second derivative uh will be negative. So yeah, when the graph is a concave upwards, the second derivative for these X values, F double prime effects for these X values will be positive. Where the graph is concave downwards, the X values, the second derivative of the X values will be negative. So for these X values, F double prime of X is negative. And the graph is concave down for these X values, F double prime of X is positive for these X values, F double prime of X. The second derivative is positive and the graph is concave upwards. Now this point, this red point right here, this red point is called an inflection point, it's where the graph changes con cavity. You actually would have another one over here Because at this point uh well, let's stay with this point over here. 1st this point is called an inflection point. So this is the point of inflection, it's where the con cavity changes here. We were concave up, we reached this point, then we became concave down. Then we're going to reach this other inflection point here and we're gonna start becoming concave up again. So inflection points are where the con cavity of the graph changes. So at this inflection point the graph changed from being concave up to becoming concave down. How do you locate an inflection point? At an inflection point, the second derivative. So we're still talking about second derivative like we did here, but at an inflection point the second derivative will equal zero, So at an inflection point the second derivative equal zero.