A curve has equation $y = f(x)$ The curve has a point of inflection at $x = 7$ It is given that $f'(7) = a$ and $f''(7) = b$, where $a$ and $b$ are real numbers. Identify which one of the statements below must be true. Circle your answer. $f'(7) eq 0$ $f'(7) = 0$ f''(7) eq 0$ f''(7) = 0

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A curve has equation $y = f(x)$ The curve has a point of inflection at $x = 7$ It is given that $f'(7) = a$ and $f''(7) = b$, where $a$ and $b$ are real numbers - AQA - A-Level Maths Pure - Question 2 - 2021 - Paper 2

Question 2

A-curve-has-equation-$y-=-f(x)$---The-curve-has-a-point-of-inflection-at-$x-=-7$---It-is-given-that-$f'(7)-=-a$-and-$f''(7)-=-b$,-where-$a$-and-$b$-are-real-numbers-AQA-A-Level Maths Pure-Question 2-2021-Paper 2.png

A curve has equation $y = f(x)$ The curve has a point of inflection at $x = 7$ It is given that $f'(7) = a$ and $f''(7) = b$, where $a$ and $b$ are real numbers.... show full transcript

Worked Solution & Example Answer:A curve has equation $y = f(x)$ The curve has a point of inflection at $x = 7$ It is given that $f'(7) = a$ and $f''(7) = b$, where $a$ and $b$ are real numbers - AQA - A-Level Maths Pure - Question 2 - 2021 - Paper 2

Step 1

Identify which one of the statements below must be true.

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Answer

To determine the behavior of the function at the point of inflection, we analyze the derivatives at that point. A point of inflection occurs where the second derivative is zero ( $f''(7) = 0$ ) and changes sign, but this does not necessitate that the first derivative ( $f'(7)$ ) is zero.

Thus, it is necessary to conclude that the correct statement is:

Answer: $f''(7) = 0$ .

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