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Examples of Stationary Points

Here are a few examples of stationary points, i.e. finding stationary points and the types of curves.

Example 1: Find the stationary point for the curve y = x3 – 3x2 + 3x – 3, and its type.

y = x3 – 3x2 + 3x – 3

dy/dx = 3x2 – 6x + 3

= 3(x2 – 2x + 1)

= 3(x – 1)2

dy/dx = 3(x – 1)2  = 0

doubleright x = 1

When x = 1, y = 13 – 3×12 + 3×1 – 3

= 1 – 3 + 3 – 3  =  -2

(1, -2) is a stationary point.

To determine it type, check dy/dx or y’ around x = 1.

x 0 (LHS) 1 2 (RHS)
dy/dx 3(-1)2 = 3 >0 0 3(1)2 = 3 >0
Curve Increasing Stationary Increasing

 

Since the curve is increasing on both sides of (1, -2), the curve is monotonic increasing.
And (1, -2) is a point of horizontal inflection.

horizontal point of inflection

 

Example 2: Find the stationary points on the curve f(x) = x3 – 3x + 2 and its type.

f(x) = x3 – 3x + 2

f'(x) = 3x2 – 3

For stationary point, f'(x) = 0

Therefore 3x2 – 3 = 0

x2 = 1,   x = pm 1

When x = 1, f(x) = 13 – 3×1 + 2 = 1 – 3 + 2  =  0

(1, 0) is the stationary point.

To find the type of stationary point, choose x = 0 on LHS of 1 and x = 2 on RHS

x 0 (LHS) 1 2 (RHS)
dy/dx 0 – 3<>= -3 < 0 0 3(2)2 – 3 >= 9 > 0
Curve Decreasing Stationary Increasing

 

The curve is decreasing, becomes zero, and then increases. Hence the curve has a local minimum point at (1, 0).

When x = -1, f(x) = (-1)3 – 3x-1 + 2 = -1 + 3 + 2  =  4

(-1, 4) is a stationary point.

To find the type of stationary point, choose x = -2 on LHS of 1 and x = 0 on RHS

x -2 (LHS) -1 0 (RHS)
dy/dx 3(-2)2 – 3 >= 9 > 0 0 3(0)2 – 3 = -9 < 0
Curve Increasing Stationary Decreasing

 

The curve is increasing, becomes zero, and then decreases. Hence the curve has a local maximum point and that is (-1, 4).

stationary point maximum & minimum