How can we calculate the maximum and minimum points of a function?

The derivative of a function in a point is, by definition, the slope of the tangent line to the function in that point. In a point that is a maximum or a minimum, one can easily picture that the tangent is flat: therefore, its slope is 0 and the derivative of the function in that point is 0. Hence, we can say that in all maximums or minimums of a function the derivative of that function is null. Beware that is not to say that whenever the derivative of a function is null, there is a maximum or a minimum: this is not true for the derivative can also be null in an inflection point. But if we are given a function to calculate its maximums and minimums, we should first find out its derivative (the derivative function). Secondly, we should equate it to zero and solve the subsequent equation. As a result we will have found out points where the derivative is null: those are maximums, minimums or inflection points. To determine which is which, we should try to substitute in the derivative function a value within each of the intervals defined by the null-derivative points. If the result of substituting in the derivative is positive, the original function is crescent in that interval; if the result of substituting the derivative is negative, the original function is decrescent in that interval. A null-derivative point is a maximum if in the interval just before it the original function is crescent (positive derivative) and in the interval just after it the original function is decrescent (negative derivative); a null-derivative point is a minimum if in the interval just before it the original function is decrescent (negative derivative) and in the interval just after it the original function is crescent (positive derivative). If in both intervals (before and after the null-derivative point) the original function is behaving in the same manner (crescent in both or decrescent in both), the null-derivative point is a point of inflection (a point where the curvature, this is the concavity or convexity, of the function changes).