Differentiation - optimisation

1. Differentiation - optimisation Finding and Identifying Maxima and minima Optimisation problems

2. Maxima, minima and stationary points Minimum dy dx dy dx Maximum Stationary points =0 B Terminology y B is a local maximum D is a local minimum Points where dy/dx=0 are call stationary points B & D are also known as turning points (where they gradient changes from +ve to -ve) D x x

3. The second derivative The second derivative is denoted by .. for a function p(x) it is denoted by p’’(x) d2y d2y d2y dx2 dx2 dx2 KEY POINTS It is used at a simple way of telling if a stationary point is a maximum or a minimum If is negative then the stationary point is a maximum If is positive then the stationary point is a minimum

4. Example d2A dx2 dA dx dA dx x x 60 - 2x Stationary points (maxima and minima) when = -4 = 60 - 4x = 0 The farmer has 60m to try and make the maximum area with a hedge as one border Hedge Area = 15 x 30 = 450m2 Area (A) = l x w = x(60 - 2x) = 60x - 2x2 60 - 4x = 0 Is this a maximum? 60 = 4x 15 = x x = 15m Which is -ve, so P is a maximum

5. Optimisation Problems - trying to get the best result by finding a maximum or a minimum - applies the methods to practical contexts For example maximising profit Minimising losses