2015 AP Calculus AB 5b | AP Calculus AB solved exams | AP Calculus AB | Khan Academy

Part B on what open intervals contained in -3 is less than x is less than four is the graph of F both concave down and decreasing. Give a reason for your answer.

All right, so F needs to be both concave down and decreasing. So, F concave down. Concave down, that's going to be true if and only if the second derivative of F is less than zero, which is true if and only if F prime is decreasing. F prime is decreasing. So, that's how we can tell when F is concave down.

And I wanted to see what do I need to observe on F prime because that's the graph they give us. All right, now for F to be decreasing, F is going to be decreasing if and only if, well, you're decreasing when you have a negative slope. If your rate of change is negative, so that's true if and only if F prime of X is less than zero.

So both true, both true when F prime of X is less than zero and decreasing. And let me write, and F prime is decreasing, and F prime decreasing. So let's see if we can identify that. Where? Let’s go back to our graph and see when is F prime of X less than zero and decreasing.

So, if we look at less than zero, let me highlight this in a, so less than zero. I'm highlighting it in magenta right over here. This is all the places where we are less than zero in magenta. And now, of that, where are we decreasing? Well, we are decreasing over this interval right over here.

So that would be -2 is less than X, which is less than -1, so that's where we're decreasing of the places where we are negative. I'm taking the subset of where we are negative to think about, well, where are we decreasing?

We're decreasing right there and we're also decreasing, we're also decreasing right over there. So that is one is less than X, which is less than three. Now, we are decreasing before X at -2 over here, but we're not less than zero there, so I'm not going to include that.

And so we have our intervals. So our open intervals, so we have both true when for when these two things are true. And so this happens when -2 is less than X is less than 1 or 1 is less than X is less than 3, so those are our two open intervals, this one and that one.