AQA A Level Maths: Pure复习笔记7.2.4 Stationary Points & Turning Points

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)