William Anthony Granville

“ A great many practical problems occur where we have to deal with functions of such a nature that they have a greatest (maximum) value or a least (minimum) value, [1] and it is very important to know what particular value of the variable gives such a value of the function. For instance, suppose that it is required to find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius 5 inches. ”

“ Maximum and minimum values of a function. A maximum value of a function is one that is greater than any values immediately preceding or following.
A minimum value of a function is one that is less than any values immediately preceding or following. ”

“ Let the base CD (= x) increase to 10 inches (the diameter) ; then the altitude will decrease to zero and the area will become zero. Now let the base decrease to zero; then the altitude will increase to 10 inches and the area will again become zero. It is therefore intuitionally evident that there exists a greatest rectangle. By a careful study of the figure we might suspect that when the rectangle becomes a square its area would be the greatest, but this would at best be mere guesswork. A better way would evidently be to plot the graph of the function (1) and note its behavior. ”