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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 $f(x)$ is defined by $f... 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 such a cubic function, we can consider a function like , which has three distinct real roots. The graph will cross the x-axis at three points and will rise to the right, ensuring that the coefficient of is positive. The sketch should show the function crossing the x-axis at the roots clearly.
Step 2
Show that there is a turning point where the curve crosses the $y$-axis.
Answer
To find the turning points, we first differentiate the function: Setting the first derivative to zero gives: Factoring this results in: Thus, we have two potential turning points:
Since 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.
Step 3
The equation $f(x) = 0$ has three distinct real roots. By considering the positions of the turning points find, in terms of $p$, the range of possible values of $q$.
Answer
To have three distinct real roots, the turning points must straddle the x-axis, meaning that:
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Evaluate the function at the turning points:
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For three distinct roots, we need: This implies: .
Thus, the range of possible values of is: .