Writing the volume in mathematical notation, we have
To maximize , we note that at the boundary points of its domain and that is differentiable everywhere, so that a maximum must be attained at a point where .
An open box is to be made from a square sheet of tin 14 centimeters on a side by cutting small squares from each of the corners and turning up the edges. What are the dimensions of the resulting box if its volume is to be maximized? Write an appropriate function modelling this situation and use the derivative to locate critical points.