1. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 7.5 Maximum and minimum points

2. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Maximum and minimum points You will already be familiar with the fact that the gradient of a curve is not constant. local maximum local minimum At two points on the curve above the gradient is zero. This occurs where the lines, drawn at a tangent to the curve at those points, are horizontal. These points are known as local maximum or minimum points.

3. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Maximum and minimum points To find the gradient function f ′(x) of any function f(x), it must be differentiated. The gradient function f ′(x) gives the equation for the gradient at all points on the curve in terms of x. f′(x) f(x) If the gradient function is plotted for the above function, we can see the relationship between the function and its gradient. Importantly, the points where the gradient function f ′(x) intercept the x-axis, have the same x-coordinates as the maximum and minimum points.

4. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Maximum and minimum points Therefore in order to find the x-coordinate of any maximum or minimum points, we solve the equation