Stationary points and increasing functions

Increasing and decreasing

• f is increasing where f'(x) > 0
• f is decreasing where f'(x) < 0

Solve the inequality to find the interval.

Stationary points

At a stationary point dy/dx = 0. Solve dy/dx = 0 to find x, then substitute into y for the coordinates.

Nature: the second derivative test

• d^2y/dx² > 0: minimum
• d^2y/dx² < 0: maximum
• d^2y/dx² = 0: the test fails. Check the gradient just either side instead (for y = x⁴ it goes from negative to positive, a minimum).

Worked example

Find and classify the stationary points of y = x³ - 3x² - 9x + 5.

1. dy/dx = 3x² - 6x - 9 = 3(x - 3)(x + 1) = 0, so x = -1 or x = 3
2. x = -1: y = -1 - 3 + 9 + 5 = 10; x = 3: y = 27 - 27 - 27 + 5 = -22
3. d^2y/dx² = 6x - 6: -12 at x = -1, 12 at x = 3

Answer: Maximum (-1, 10); minimum (3, -22)

Check yourself

1. Find the x-coordinates of the stationary points of y = x³ - 12x + 1.
   Show answer

   x = 2 or x = -2. dy/dx = 3x² - 12 = 0, so x² = 4.

2. Find the y-coordinate of the minimum point of y = x² - 8x + 3.
   Show answer

   -13. dy/dx = 2x - 8 = 0 at x = 4, and y = 16 - 32 + 3 = -13.

3. y = x³ - 12x + 1 has a stationary point at x = 2. What is its nature?
   • A maximum, because d^2y/dx² = 12 > 0
   • A point of inflection
   • It cannot be decided
   • A minimum, because d^2y/dx² = 12 > 0
   Show answer

   A minimum, because d^2y/dx² = 12 > 0. d^2y/dx² = 6x = 12 at x = 2. Positive means a minimum.

4. Which is true for f(x) = x³ + 3x?
   • f is increasing for all x
   • f is decreasing for all x
   • f has two stationary points
   • f is increasing only for x > 0
   Show answer

   f is increasing for all x. f'(x) = 3x² + 3, which is always positive.

5. For which values of x is y = x² - 6x decreasing?
   • x > 3
   • x < 0
   • x < 3
   • 0 < x < 6
   Show answer

   x < 3. dy/dx = 2x - 6 < 0 when x < 3.

6. For y = x⁴, dy/dx = 0 and d^2y/dx² = 0 at x = 0. What is the nature of this point?
   • A minimum: the gradient changes from negative to positive
   • A point of inflection, because d^2y/dx² = 0
   • A maximum
   • It is not a stationary point
   Show answer

   A minimum: the gradient changes from negative to positive. The test fails, so check either side: dy/dx = 4x³ is negative for x < 0 and positive for x > 0.

7. f(x) = x³ - 6x² + 9x is decreasing for a < x < b. Find b.
   Show answer

   3. f'(x) = 3x² - 12x + 9 = 3(x - 1)(x - 3) < 0 for 1 < x < 3, so b = 3.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.