Finding Stationary Points

Finding stationary points Step 1 Starting with 𝑦 = 𝑓 (𝑥) differentiate it twice to get

𝑑𝑦 𝑑𝑥

and

𝑑2 𝑦 𝑑𝑥 2

.

Step 2 𝑑𝑦

Solve

𝑑𝑥

=0.

Step 3 𝑑2 𝑦

For each x value found in Step 2, find the value of y, and the sign of 𝑑𝑥 2 (find whether the second derivative is greater than zero, less than zero or equal to zero). Step 4 If If If

𝑑2 𝑦 𝑑𝑥 2 𝑑2 𝑦 𝑑𝑥 2 𝑑2 𝑦 𝑑𝑥 2

> 0, you have a minimum point. < 0, you have a maximum point. = 0, then it could be a minimum, maximum or a point of inflection. Go to Step 5.

Otherwise, go to Step 6. Step 5 𝑑3 𝑦

Find the third derivative: 𝑑𝑥 3 . If If

𝑑2 𝑦 𝑑𝑥 2 𝑑2 𝑦 𝑑𝑥 2

= 0 and

𝑑3 𝑦 𝑑𝑥 3

≠ 0, then you have a point of inflection.

𝑑3 𝑦

= 0 and𝑑𝑥 3 = 0, then check the gradient either side of the x value.

Step 6 If required, sketch the curve. Remember: a. the first derivative tells us where the stationary point is; b. the second (and possibly third) derivative tells us what type the stationary point is.

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