A cubic function, $f(x)$, is defined on the set of real numbers - Scottish Highers Maths - Question 15 - 2018
Question 15
A cubic function, $f(x)$, is defined on the set of real numbers.
• $(x + 4)$ is a factor of $f(x)$
• $x = 2$ is a repeated root of $f(x)$
• $f'(-2) = 0$
• $f'... show full transcript
Worked Solution & Example Answer:A cubic function, $f(x)$, is defined on the set of real numbers - Scottish Highers Maths - Question 15 - 2018
Step 1
• $(x + 4)$ is a factor of $f(x)$
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Answer
This indicates that f(−4)=0. Therefore, the graph must cross the x-axis at x=−4.
Step 2
• $x = 2$ is a repeated root of $f(x)$
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Answer
Since x=2 is a repeated root, the graph must touch the x-axis at this point but not cross it. Therefore, it will have a stationary point here.
Step 3
• $f'(-2) = 0$
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At x=−2, there is also a stationary point identifiable from the graph. The graph should show a local maximum or minimum at this point.
Step 4
• $f'(x) > 0$ where the graph with equation $y = f(x)$ crosses the y-axis
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This means that to the left of the y-axis, the function is increasing. The sketch should show the graph coming from below at the y-axis and rising to the right.
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