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 $f(x) = x^3 + 3px^2 + q$ 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 Applications of Differentiation: 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 $f(x)$ 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 with three distinct real roots, we can take a generic form of the function, such as:

$f(x) = (x - r_1)(x - r_2)(x - r_3)$

where $r_1$ , $r_2$ , and $r_3$ are the three distinct real roots. Since we want the leading coefficient to be positive, we ensure that $f(x)$ opens upwards. For example, choosing $r_1 = -2$ , $r_2 = 0$ , and $r_3 = 1$ gives:

$f(x) = (x + 2)(x)(x - 1) = x^3 + x^2 - 2x$

The sketch will show the curve crossing the $x$ -axis at $x = -2$ , $x = 0$ , and $x = 1$ , thus confirming three distinct real roots.

Step 2

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

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Answer

First, we differentiate the function:

$f'(x) = 3x^2 + 6px$

Setting this equal to zero for turning points:

3x(x + 2p) = 0$$ This gives: 1. $x = 0$ 2. $x = -2p$ Since $p > 0$, $-2p < 0$. Therefore, $x = 0$ is indeed a turning point where the curve crosses the $y$-axis, which confirms the presence of 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 $q$.

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Answer

We have already defined the turning points as $x = 0$ and $x = -2p$ . To have three distinct real roots, we require:

The turning point at $x = -2p$ $x = - 2 p$ must be a local maximum, and therefore:
- $f(-2p)$ should be positive, and $f(0)$ must be negative for a change in sign (indicating distinct roots).

Calculating:

$f(0) = q$
$f(-2p) = (-2p)^3 + 3p(-2p)^2 + q = -8p^3 + 12p^3 + q = 4p^3 + q$

Thus, we must have:

$f(0) < 0 ext{ and } f(-2p) > 0$
which leads to:

$q < 0 ext{ and } 4p^3 + q > 0$
or $-4p^3 < q < 0$

This provides the range of possible values for $q$ in terms of $p$ .

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. By considering the positions of the turning points, find, in terms of $p$, the range of possible values of $q$.

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