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 f'(x) at x = 0
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Answer
We see that f'(x) changes from positive to zero at x = 0, indicating a local maximum at this point. This suggests that the graph will have a peak at x = 0.
Step 2
Analyze f''(x) at x = 0
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Answer
The value of f''(x) at x = 0 is zero. However, since f''(x) is negative for x < 0, this indicates that the function is concave down before x = 0, reinforcing that there is a maximum at this point.
Step 3
Consider f'(x) at x = -2
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Answer
As f'(x) is positive for x < 0, the function is increasing before reaching x = 0.
Step 4
Consider f''(x) at x = 2
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At x = 2, f'(x) is positive and f''(x) is positive as well. This indicates that the function is both increasing and concave up after x = 0.
Step 5
Conclusion on the possible sketch
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Answer
Based on the analysis above, sketch C reflects the changes in increasing and decreasing behavior along with concavity, making it the correct choice. The graph increases to a maximum at x = 0 and then continues increasing after x = 0.
