Classifying stationary points

What does the derivative mean?

• If , the function is increasing (as x increases, y increases)
• If , the function is decreasing (as x increases, y decreases)
• If , the function is stationary

What does the second derivative mean?

If at some point , then we can use the second derivative to find the type of stationary point:

• If , the point is a local minimum.
• If , the point is a local maximum.
• If , it could be a point of inflection, but may not be. We can’t say for sure what type of stationary point it is.

Example: find the stationary points of and classify each

• Set :
  • or
• Substitute back to find y-coordinates:
  • At : -> Stationary point at (0, 4)
  • At : -> Stationary point at (2, 0)
• Now we find the second derivative:
• Evaluate the second derivative at each stationary point:
  • At : (less than 0) -> Local maximum at (0, 4)
  • At : (greater than 0) -> Local minimum at (2, 0)
• Answer: Local maximum at (0, 4) and local minimum at (2, 0)

Example: find the stationary points of and classify each

• Set :
  • or
  • or
  • (the other two solutions are complex numbers)
• Substitute back to find y-coordinate:
  • At : -> Stationary point at (0, 0)
• Now we find the second derivative:
• Evaluate the second derivative at the stationary point:
  • At : (greater than 0) -> Local minimum at (0, 0)
• Answer: Local minimum at (0, 0)