maximum and minimum

A maximum point on a curve is higher than the points immediately on either side of it; it is not necessarily the highest point on the curve. Similarly, a minimum point is lower than the points immediately on either side.

In coordinate geometry, points at which the slope of a curve representing a function changes from positive to negative (maximum), or from negative to positive (minimum). A tangent to the curve at a maximum or minimum has zero gradient.

Maxima and minima can be found by differentiating the function for the curve and setting the differential to zero (the value of the slope at the turning point). For example, differentiating the function for the parabola y = 2x² − 8x gives dy/dx = 4x − 8. Setting this equal to zero gives x = 2, so that y = −8 (found by substituting x = 2 into the parabola equation). Thus the function has a minimum at the point (2, −8).