The graph of $y = f(x)$ is shown - HSC - SSCE Mathematics Advanced - Question 10 - 2020 - Paper 1
Question 10
The graph of $y = f(x)$ is shown.
Which of the following inequalities is correct?
A. $f''(1) < 0 < f'(1)$
B. $f''(1) < 0 < f'(1)$
C. $0 < f''(1) < f'(1)$
D... show full transcript
Worked Solution & Example Answer:The graph of $y = f(x)$ is shown - HSC - SSCE Mathematics Advanced - Question 10 - 2020 - Paper 1
Step 1
Identify the behavior of the function at $x = 1$
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Answer
To analyze the behavior of the function at the point x=1, we can observe the curve on the graph. The function f(x) appears to be increasing as it passes through the point (1,f(1)), indicating that f′(1)>0.
Step 2
Examine the concavity at $x = 1$
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Answer
At x=1, the function is concave down as the curve is bending downwards. This suggests that the second derivative f′′(1) is negative, therefore f′′(1)<0.
Step 3
Combine findings to select the correct inequality
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Answer
Based on the findings from the first two steps, we found that f′(1)>0 and f′′(1)<0. Thus, we can conclude that the correct inequality is 0<f′′(1)<f′(1), aligning with option D.
