0Find the stationary point for the curve y = x3 – 3x2 + 3x – 3, and its type.
y = x3 – 3x2 + 3x – 3
= 3x2 – 6x + 3
= 3(x2 – 2x + 1)
= 3(x – 1)2
= 3(x – 1)2 = 0
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 or y’ around x = 1.
| x | 0 (LHS) | 1 | 2 (RHS) |
|
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.
