Based on the graph f(x)f\left(x\right)f(x), describe all points where the derivative f′(x)f^{\prime}\left(x\right)f′(x) would have a jump.

x = − 1 x=-1 x = − 1 and x = 0.5 x=0.5 x = 0.5

Based on the graph f ( x ) f\left(x\right) f ( x ) , describe all points where the derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) would have a jump.

So in this problem, we're told based on the graph, f of x, describe all the points where the derivative f prime of x would have a jump. So this would, in essence, mean there's a discontinuity in the derivative graph, and recall that this happens whenever the function is discontinuous at a point or has a sharp turn. Now looking at this function here, I can trace this whole function without picking up the pen. And because of that, that means that this is continuous everywhere, but it's not necessarily going to have a derivative that exists everywhere because we need to watch out for sharp turns or corners as well. Well, I can see that there's a sharp corner that appears right about here, and I can see that there's a sharp corner that appears right there. So what that means is at x equals negative one and at x equals positive 0.5 or 1 half, there is going to be a jump in our derivative graph, and that is going to be the solution to this problem. So I hope you found this helpful, and let's move on.