“Teach A Level Maths” Vol. 1: AS Core Modules 28: Harder Stationary Points © Christine Crisp

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The stationary points of a curve are the points where the gradient is zero We may be able to determine the nature of a stationary point just by knowing the shape of a curve. e.g.1 We know the curve has a minimum because the sign of the term ( positive ) tells us that the graph has the following shape

e.g.1 The cubic curve has 2 stationary points. They are and Plotting the stationary points and using our knowledge that a cubic is a continuous function ( we can draw it with a single stroke ) means we must get the following: x x The y-intercept is also useful

e.g.1 The cubic curve has 2 stationary points. They are and Plotting the stationary points and using our knowledge that a cubic is a continuous function ( we can draw it with a single stroke ) means we must get the following: x

We may not know the shape of some functions, so we need to determine the nature of the stationary points by another method. Using the 2nd derivative is usually the easiest method.

Distinguish between the max and the min. e.g.2 Calculate the coordinates of the stationary points on the graph of where Distinguish between the max and the min. must be written in the form before we can differentiate Solution: For st. pts. Multiply by : this quadratic equation has no linear term so there is no need to factorize

N.B. The maximum has a smaller y -value than the minimum ! Calculate y-values at x = 1 and -1: The stationary points are ( 1, 2 ) and (-1, -2) To distinguish between the stationary points we need the 2nd derivative is a min is a max N.B. The maximum has a smaller y -value than the minimum ! It’s interesting to see what the graph looks like.

x = 0 is infinite, so x = 0 (the y-axis) is an asymptote So, we now have x = 0 x (min) x (max)

“ x approaches infinity “ Also, as so “ x approaches infinity “ “ approaches x “ “ approaches zero “ is also an asymptote x = 0 y = x x (min) x (max)

We can now complete the curve. x (max) (min) Asymptote, x = 0

Exercise 1. Find the stationary points on the curve where Determine the nature of the stationary points. Ans: is a maximum is a minimum The question didn’t ask for the graph but it looks like this: The asymptotes are

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Distinguish between the max and the min. Solution: e.g.2 Calculate the coordinates of the stationary points on the graph of where Multiply by : must be written in the form before we can differentiate For st. pts. this quadratic equation has no linear term so there is no need to factorize

N.B. The maximum has a smaller y -value than the minimum ! To distinguish between the stationary points we need the 2nd derivative The stationary points are ( 1, 2 ) and (-1, -2) is a min is a max Calculate y-values at x = 1 and -1: