0580/43 May/June 2022 Question 9: Stationary Points and Turning Points

Question 1

Draw the curve y = (x + 1)(3 − x)(3 + x) and label where it meets both axes.

Accepted Answer

y = 0 when x = −1, x = 3, x = −3: crosses the x-axis at −3, −1 and 3 x = 0: y = 1 × 3 × 3 = 9, crosses the y-axis at 9 The x³ term is −x³, so draw a negative cubic (high on the left, low on the right) through these points Answer: Correct sketch of negative cubic crossing the x-axis at −3, −1 and 3 and crossing the y-axis at 9

Question 2

Expand the product of the three brackets to obtain y = 9 + 9x − x² − x³.

Accepted Answer

(3 − x)(3 + x) = 9 − x² (x + 1)(9 − x²) = 9x − x³ + 9 − x² y = 9 + 9x − x² − x³ Answer: Correct completion to y = 9 + 9x − x² − x³

Question 3

Find the x-coordinates of the stationary points of y = 9 + 9x − x² − x³, showing working, to 2 decimal places.

Accepted Answer

dy/dx = 9 − 2x − 3x² 9 − 2x − 3x² = 0, so 3x² + 2x − 9 = 0 x = (−2 ± √(2² − 4 × 3 × (−9)))/(2 × 3) = (−2 ± √112)/6 x = −2.097… or x = 1.430… x = −2.10 and x = 1.43 Answer: −2.10 and 1.43

Question 4

For 9 + 9x − x² − x³ = k having exactly one root, determine the largest integer a and smallest integer b with k < a or k > b.

Accepted Answer

At x = −2.097…: y = −5.049… (minimum) At x = 1.430…: y = 16.90… (maximum) One solution when k < −5.049… or k > 16.90… a = −6, b = 17 Answer: a = −6, b = 17

0580/43 May/Jun 2022, Question 9: Stationary Points and Turning Points

The question, in short

Part (a) · 4 marks

Part (bi) · 2 marks

Part (bii) · 7 marks

Part (biii) · 3 marks

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