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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
Question 7
7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$. 7 (b) The function $f(t)$ is defined by ... show full transcript
Worked Solution & Example Answer: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
Step 1
Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$
Answer
To sketch the graph of a cubic function that meets these criteria, one can consider a function such as . This function will have three distinct real roots where it crosses the x-axis at three points. The graph should show an increase in the positive direction as approaches positive infinity, and a decrease in the negative direction as approaches negative infinity, characteristic of cubic functions with a positive leading coefficient.
Step 2
Show that there is a turning point where the curve crosses the $y$-axis.
Answer
To find the turning points of the function, we first differentiate it: Set the derivative to zero to find turning points:
3x(x + 2p) = 0.$$ This yields $x = 0$ or $x = -2p$. Since $p > 0$, $-2p < 0$ which confirms that there is a turning point at $x = 0$, where the curve crosses the y-axis.Step 3
The equation $f(x) = 0$ has three distinct real roots.
Answer
To ensure that the function has three distinct real roots, we need to analyze the positions of turning points. The turning point at is a maximum and the one at is a minimum due to the nature of the cubic function. Evaluating the function at these points gives us: and For the cubic to cross the x-axis three times, the max at , which is , must be greater than 0, while the minimum at must be less than 0:
\ q < 4p^3.$$ Thus, the range for $q$ is: $$-4p^3 < q < 0.$$