Calculus: turning points, maxima, minima and motion

International GCSE Higher uses differentiation to find turning points, to solve 'largest' and 'smallest' problems, and to describe motion.

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Turning points

At a turning point the gradient is 0, so solve dydx = 0.
y = x3 − 12x: dydx = 3x2 − 12 = 0, so x2 = 4 and x = −2 or 2.
y-values: x = −2 gives y = 16; x = 2 gives y = −16.

Maximum or minimum?

Use the shape of the graph. A positive cubic rises, falls, then rises again, so the left turning point is a maximum and the right one a minimum.
You can also check the gradient either side: + then − is a maximum; − then + is a minimum.
For y = x3 − 12x: (−2, 16) is a maximum and (2, −16) a minimum.

Motion and practical problems

Displacement s, velocity v = dsdt, acceleration a = dvdt.
s = t3 − 4t2 + 5t: v = 3t2 − 8t + 5 and a = 6t − 8.
For a practical problem, write the quantity in one variable, differentiate and set equal to 0.
A = x(40 − 2x): dAdx = 40 − 4x = 0, so x = 10 and A = 200.
Worked example

s = t3 − 4t2 + 5t. Find the velocity when t = 2.

  1. v = dsdt = 3t2 − 8t + 5
  2. t = 2: 12 − 16 + 5 = 1

Answer: v = 1

Key idea

Solve dy/dx = 0 for turning points; decide maximum or minimum from the shape or the gradient either side. Velocity = ds/dt and acceleration = dv/dt.

Key words
Gradient
The steepness of a line: change in y ÷ change in x. It is m in y = mx + c.
Turning point
The point where a curve changes from going down to going up or the other way: a minimum or maximum.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. Find the y-coordinate of the maximum point of y = x3 − 27x.

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    Answer: 54

    3x2 − 27 = 0 gives x = ±3. The maximum is at x = −3: (−3)3 − 27(−3) = −27 + 81 = 54.

  2. y = x3 − 12x. What kind of turning point is at x = −2?

    1. It is not a turning point
    2. Neither
    3. A minimum
    4. A maximum
    Show the answer

    Answer: A maximum

    dydx = 3x2 − 12 = 0 at x = −2. d2ydx2 = 6x = −12, negative, so a maximum.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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