7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ - AQA - A-Level Maths Pure - Question 7 - 2019 - Paper 2

To find the turning points, we first differentiate the function: $f'(x) = 3x^2 + 6px$ Setting the first derivative to zero gives: $3x^2 + 6px = 0$ Factoring this results in: $3x(x + 2p) = 0$ Thus, we have two potential turning points:

• $x = 0$
• $x = -2p$

Since $x = 0$ is one of the turning points, and it is where the curve crosses the y-axis, we conclude that there is indeed a turning point at the y-axis.

To have three distinct real roots, the turning points must straddle the x-axis, meaning that:

1. Evaluate the function at the turning points: $f(0) = 0^3 + 3p(0^2) + q = q$ $f(-2p) = (-2p)^3 + 3p(-2p)^2 + q = -8p^3 + 12p^3 + q = 4p^3 + q$

2. For three distinct roots, we need: $q < 0$ $4p^3 + q > 0$ This implies: $-4p^3 < q < 0$.

Thus, the range of possible values of $q$ is: $-4p^3 < q < 0$.