19M.2.AHL.TZ2.H_9

pestleMathematicsAAHLPaper 219M· sl-5-8-testing-for-max-and-min-optimisation-points-of-inflexionsource ↗

Consider the polynomial $P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}$ .

Sketch the graph of $y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51$ , stating clearly the coordinates of any maximum and minimum points and intersections with axes.

[6]

Hence, or otherwise, state the condition on $k \in \mathbb{R}$ such that all roots of the equation $P\left( z \right) = k$ are real.

[2]

Markscheme / solution

shape A1

$x$ -axis intercepts at (−3, 0), (1, 0) and $y$ -axis intercept at (0, −51) A1A1

minimum points at (−1.62, −118) and (3.72, 19.7) A1A1

maximum point at (2.40, 26.9) A1

Note: Coordinates may be seen on the graph or elsewhere.

Note: Accept −3, 1 and −51 marked on the axes.

[6 marks]

from graph, 19.7 ≤ $k$ ≤ 26.9 A1A1

Note: Award A1 for correct endpoints and A1 for correct inequalities.

[2 marks]

Examiners’ report

[N/A]