So this next one says, so we have the exact same setup, which actually all of these examples will have, g of x is equal to this thing here. What is an appropriate calculus-based justification for the fact that g has a relative minimum at x equals eight? So once again, they've graphed f here, which is the same thing as the derivative of g, and so if we have the graph of the derivative, how can we say, how do we know that we have a relative minimum at x equals eight? Well, if the fact that we cross, that we're at the x-axis, that x equals, that y is equal to zero, that the function, that the derivative is equal to zero at x equals eight, that tells us that the slope of the tangent line of g at that point is zero, but that alone does not tell us we have a relative minimum point. In order to have a relative minimum point, our derivative has to cross from being negative to positive. Why is that valuable? Because think about, if your derivative goes from being negative to positive, that means your original function goes from decreasing to increasing, goes from decreasing to increasing, and so you would have a relative minimum point.