Graphs 9
2 marks · EasyConsider the cubic function . A stationary point occurs when the gradient is zero.
State why the gradient at a stationary point is zero.
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At a stationary point, the tangent to the curve is horizontal therefore the gradient is zero.
Also accept: Because at a stationary point the curve does not rise or fall, so its gradient is zero.; Because the slope of the curve is zero at a stationary point.; The gradient is zero at a stationary point since the curve changes from increasing to decreasing (or vice versa).
Common mistakes: Saying the gradient is zero without mentioning the tangent is horizontal.; Confusing gradient zero with maximum/minimum without explanation.
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Marking points
B1 (a) · 1 mark · States that at a stationary point, the gradient or slope is zero.
B2 (a) · 1 mark · Correctly links this to the idea of a horizontal tangent or the curve not rising or falling.
Worked solution
A stationary point is where the gradient (slope) of the curve becomes zero.
This means the tangent to the curve at that point is horizontal, so the curve is not rising or falling there.