What are stationary points?

• A stationary point is any point on a curve where the gradient is zero
• To find stationary points of a function?f(x)

Step 1:?Find the first derivative?f'(x)

Step 2:?Solve?f'(x)?=?0?to find the?x-coordinates of the stationary points

Step 3:?Substitute those?x-coordinates into?f(x)?to find the corresponding?y-coordinates

• A stationary point may be either a?local minimum, a?local maximum, or a?point of inflection

Stationary points on quadratics

• The graph of a quadratic function (ie a parabola) only has a single stationary point
• For an 'up' parabola this is the?minimum; for a 'down' parabola it is the?maximum?(no need to talk about 'local' here)

• The?y?value of the stationary point is thus the minimum or maximum value of the quadratic function
• For quadratics especially minimum and maximum points are often referred to as?turning points

How do I determine the nature of stationary points on other curves?

• For a function?f(x)?there are two ways to determine the nature of its stationary points?
• Method A:?Compare the signs of the?first derivative?(positive or negative) a little bit?to either side of the stationary point
  • (After completing?Steps 1 - 3 above?to find the stationary points)

Step 4:?For each stationary point find the values of the first derivative a little bit 'to the left' (ie slightly smaller?x?value) and a little bit 'to the right' (slightly larger?x?value) of the stationary point

Step 5:?Compare the?signs?(positive or negative) of the derivatives on the left and right of the stationary point

• • If the derivatives are?negative?on the?left?and?positive?on the?right, the point is a?local minimum
  • If the derivatives are?positive?on the?left?and?negative?on the?right, the point is a?local maximum
  • If the signs of the derivatives are the?same?on both sides (both positive or both negative) then the point is a?point of inflection

   

Method B:?Look at the sign of the?second derivative?(positive or negative)?at the stationary point

(After completing?Steps 1 - 3 above?to find the stationary points)

Step 4:?Find the second derivative?f''(x)

Step 5:?For each stationary point find the value of?f''(x)?at the stationary point (ie substitute the?x-coordinate of the stationary point into?f''(x)?)

• • If?f''(x)?is?positive?then the point is a?local minimum
  • If?f''(x)?is?negative?then the point is a?local maximum
  • If?f''(x)?is?zero?then the point could be a?local minimum, a?local maximum?OR a?point of inflection?(use Method A to determine which)