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.