Function Minima and Maxima
Prerequisites
Cartesian Coordinates Show
Gradient of a straight line Show
Equation of a straight line Show
Function Minima and Maxima
Introduction
- Given a function:
- We can draw it as a graph:
- We see that the graph has two turning points:
- These are called the local minima and maxima.
- They are the points at which the function reaches a minimum or maximum.
- They are also called the "turning points" of the function.
Finding the minima and maxima
- First we find the derivative of the function:
- If we plot this on the same graph:
- The blue line is our original function.
- The red line is it's derivative.
- We see that at the minima and maxima, the derivative is 0.
- The derivative is the gradient of the function.
- At the turning points, the gradient is 0.
- The points where the function is 0 are called the functions roots:
- x = -1.215... x = 0.549...
- Finding the roots of a function is a huge subject that I wont cover here.
- Finally, we can use these x values in our original equation to find the coordinates:
(-1.215, 2.113), (0.549, -0.631)
Checking if it's minimum or maximum
- Given the location of a turning point, we can determine if it's a minimum or maximum.
- to start, we calculate the second derivative of the function:
- And graphing it:
- The blue line is our original equation.
- The red line is the first derivative.
- The green line is the second derivative.
- For our turning point at (-1.215, 2.113)
- It's a maximum.
- The derivative goes from positive to negative.
- It's derivative's gradient (second derivative) is -5.29 (at x = -1.215)
- for our turning point at (0.549, -0.631)
- It's a minimum.
- The derivative goes from negative to positive.
- It's derivative's gradient (second derivative) is 5.294 (at x = 0.549)
- We can therefore create the rule:
- If the second derivative is negative at x then the turning point is a maximum.
- If the second derivative is positive at x then the turning point is a minumum.