Finding stationary points
- A stationary point is where the gradient is zero: dy/dx = 0.
- Method: differentiate, set dy/dx = 0, solve for x, then substitute into y to find the coordinates.
- y = x² − 4x + 1: dy/dx = 2x − 4 = 0, so x = 2 and y = −3. The stationary point is (2, −3).
Maximum or minimum?
- Check the gradient either side. Positive, then 0, then negative means a maximum. Negative, then 0, then positive means a minimum.
- Or use the second derivative, d²y/dx²: positive means a minimum, negative means a maximum.
- A stationary point can also be a point of inflection, where the curve flattens but keeps going the same way (such as y = x³ at the origin).
Worked example
- y = x³ − 3x² − 9x + 5. dy/dx = 3x² − 6x − 9 = 3(x − 3)(x + 1), so x = 3 or x = −1.
- At x = 3: y = 27 − 27 − 27 + 5 = −22. At x = −1: y = −1 − 3 + 9 + 5 = 10.
- d²y/dx² = 6x − 6. At x = 3 it is 12 (positive), so (3, −22) is a minimum. At x = −1 it is −12 (negative), so (−1, 10) is a maximum.
Increasing and decreasing
- A function is increasing where dy/dx > 0. y = x² − 4x + 1 is increasing for x > 2, and decreasing for x < 2.
Key terms
- Stationary point
- A point where the gradient is zero.
- Maximum point
- A stationary point where the curve changes from going up to going down.
- Minimum point
- A stationary point where the curve changes from going down to going up.
- Point of inflection
- A stationary point where the curve doesn't change direction.
- Second derivative
- The result of differentiating dy/dx again, written d²y/dx².