What is an appropriate calculus-based justification for the fact that g is concave up on the interval, the open interval from five to 10? So concave up. So before I even think about what it means to be concave up, let's just make sure we understand this relationship between g and f. One way to understand it is if we took the derivative of both sides of this equation, we would get that g prime of x is equal to f of x. The derivative of this with respect to x would just be f of x. In fact, the whole reason why we introduce this variable t here is this thing right over here is actually a function of x, because x is this upper bound. And it would've been weird if we had x as an upper bound, or at least confusing, and we're also integrating with respect to x. So we just have to pick kind of another placeholder variable that doesn't have to be t. It could be alpha, it could be gamma, it could be a, b, or c, whatever we choose.