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The diagram shows the graph of $y = f(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2016 - Paper 1

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Question 9

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The diagram shows the graph of $y = f(x)$. Which of the following is a correct statement? (A) $f''(1) < f(1) < 1 < f'(1)$ (B) $f''(1) < f'(1) < f(1) < 1$ ... show full transcript

Worked Solution & Example Answer:The diagram shows the graph of $y = f(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2016 - Paper 1

Step 1

Identify the Function and Its Derivatives

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Answer

From the graph provided, we can analyze the behavior of the function f(x)f(x) at the point x=1x=1.

  1. The function appears to reach its maximum at this point, suggesting that f′(1)=0f'(1) = 0.
  2. At a maximum, the second derivative must be negative, indicating that f′′(1)<0f''(1) < 0.

Step 2

Evaluate Each Option

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Answer

Let's evaluate each option:

  • Option (A): f′′(1)<f(1)<1<f′(1)f''(1) < f(1) < 1 < f'(1)

    • Since f′(1)=0f'(1) = 0, this cannot be true.

  • Option (B): f′′(1)<f′(1)<f(1)<1f''(1) < f'(1) < f(1) < 1

    • Since f′′(1)<0f''(1) < 0 and f′(1)=0f'(1) = 0, this holds: f′′(1)<f′(1)<f(1)<1f''(1) < f'(1) < f(1) < 1.

  • Option (C): f(1)<1<f′′(1)<f′(1)f(1) < 1 < f''(1) < f'(1)

    • This is incorrect as f′′(1)<0f''(1) < 0 and f′(1)=0f'(1) = 0.

  • Option (D): f′(1)<f(1)<1<f′′(1)f'(1) < f(1) < 1 < f''(1)

    • This is also incorrect because f′′(1)<0f''(1) < 0.

Thus, only option (B) is valid.

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