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 $$f(t) = x^3 + 3px^2 + q$$ where $p$ and $q$ are constants and $p > 0$. 7 (b) (i) Show that there is a turning point where the curve crosses the $y$-axis. 7 (b) (ii) 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 $q$.

A-Level AQA Maths Pure Polynomials: 7 (a) Sketch the graph of any cu

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-$x^3$-AQA-A-Level Maths Pure-Question 7-2019-Paper 2.png

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 $x^3$

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Answer

To sketch the graph of a cubic function that meets these criteria, one can consider a function such as $f(x) = x^3 - 3x + 2$ . 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 $x$ approaches positive infinity, and a decrease in the negative direction as $x$ 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.

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Answer

To find the turning points of the function, we first differentiate it: $f'(x) = 3x^2 + 6px.$ 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.

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Answer

To ensure that the function has three distinct real roots, we need to analyze the positions of turning points. The turning point at $x = 0$ is a maximum and the one at $x = -2p$ is a minimum due to the nature of the cubic function. Evaluating the function at these points gives us: $f(0) = q$ and $f(-2p) = -4p^3 + q.$ For the cubic to cross the x-axis three times, the max at $x = 0$ , which is $f(0) = q$ , must be greater than 0, while the minimum at $x = -2p$ must be less than 0:

\ q < 4p^3.$$ Thus, the range for $q$ is: $$-4p^3 < q < 0.$$

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

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

Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$

Show that there is a turning point where the curve crosses the $y$-axis.

The equation $f(x) = 0$ has three distinct real roots.

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