The diagram shows the graph $y = f(x)$ - HSC - SSCE Mathematics Advanced - Question 10 - 2024 - Paper 1
Question 10
The diagram shows the graph $y = f(x)$.
The point Q is a horizontal point of inflection.
Let $A(x) = \int_0^x f(t) dt$.
How many points of inflection does the ... show full transcript
Worked Solution & Example Answer:The diagram shows the graph $y = f(x)$ - HSC - SSCE Mathematics Advanced - Question 10 - 2024 - Paper 1
Step 1
Determine Points of Inflection of $y = f(x)$
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Answer
In the graph of y=f(x), a point of inflection occurs where the concavity changes. Since point Q is specified as a horizontal point of inflection, we need to analyze the segments of the curve before and after point Q to identify any changes in concavity.
Step 2
Apply the Fundamental Theorem of Calculus
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Answer
Given that ( A(x) = \int_0^x f(t) dt ), the derivative A′(x)=f(x). Points of inflection in y=A(x) will occur where A′′(x)=f′(x)=0.
Step 3
Analyze $f'(x)$ Based on the Graph of $f(x)$
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Answer
Since Q is a horizontal point of inflection, f′(x) changes sign around Q indicating that there are also inflection points in A(x) near Q.
Step 4
Count Points of Inflection for $A(x)$
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Answer
There are a total of 3 points of inflection for the graph y=A(x), which includes the change in concavity at Q and the segments before and after it.
