The following table gives the signs of the first and second derivatives of a function $y = f(x)$ for different values of $x$ - HSC - SSCE Mathematics Advanced - Question 6 - 2023 - Paper 1
Question 6
The following table gives the signs of the first and second derivatives of a function $y = f(x)$ for different values of $x$.
| x | -2 | 0 | 2 |
|----|----|---|---... show full transcript
Worked Solution & Example Answer:The following table gives the signs of the first and second derivatives of a function $y = f(x)$ for different values of $x$ - HSC - SSCE Mathematics Advanced - Question 6 - 2023 - Paper 1
Step 1
Analyze the signs of $f'(x)$ and $f''(x)$ at $x = -2$
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Answer
At x=−2, f′(x) is positive, which indicates that the function is increasing at this point. Since f′′(x) is negative, this implies the function is concave down. Therefore, the graph is rising but will start to decrease after this point.
Step 2
Analyze the signs of $f'(x)$ and $f''(x)$ at $x = 0$
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Answer
At x=0, f′(x) is zero, indicating a critical point. The second derivative f′′(x) is also zero, suggesting the concavity changes here. This is where the function could potentially reach a local maximum or minimum.
Step 3
Analyze the signs of $f'(x)$ and $f''(x)$ at $x = 2$
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Answer
At x=2, f′(x) is positive, showing that the function is increasing. The second derivative f′′(x) is positive, which indicates the function is also concave up, implying that it continues increasing.
Step 4
Choose the correct sketch based on the analysis
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Answer
Considering the information above, the graph should increase until x=−2, have a critical point at x=0, and then continue to increase with a concave up shape after x=2. The correct sketch is therefore option C.
