The graph of g has a cup u shape on the interval. Well, if we had the graph of g, this would be a justification, but it wouldn't be a calculus-based justification. Let's do more of these. 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.