Graphs 9
4 marks · HardFind the coordinates of the stationary points of the curve .
Determine the nature (maximum or minimum) of each stationary point.
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(a) (2, 3), (4, 5)
(b) (2, 3) is a minimum, (4, 5) is a maximum
Common mistakes: Error in solving quadratic for stationary points; Incorrect second derivative or wrong classification
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Finds and sets it to zero to find stationary points.
A1 (a) · 1 mark · Correctly solves for x and finds both coordinates.
M2 (b) · 1 mark · Uses the second derivative to attempt classification.
A2 (b) · 1 mark · Correctly identifies minimum and maximum. (follow-through)
Worked solution
Differentiate:
Set :
So there is a repeated root. Find y at x = 2:
So the stationary point is (2, 3). To classify:
Because the second derivative is zero, this is a point of inflection, not a max or min.