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The figure shows graphs of f, f', f", and f". Identify each curve, and explain your choices.

The figure shows graphs of f, f', f", and f". Identify each curve, and explain your choices.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Problem 50 **Question:** The figure shows graphs of \( f \), \( f' \), \( f'' \), and \( f''' \). Identify each curve, and explain your choices. **Solution:** The given figure displays four different graphs labeled as \( a \), \( b \), \( c \), and \( d \), and we need to identify which one corresponds to the function \( f \), its first derivative \( f' \), its second derivative \( f'' \), and its third derivative \( f''' \). - **Curve \( d \) (Black curve):** This curve has inflection points and shows changes in concavity, indicating it is the original function \( f(x) \). - **Curve \( c \) (Blue curve):** This curve appears to have roots where the original function \( f(x) \) changes direction. These roots correspond to the critical points of \( f(x) \), suggesting that \( c \) is the first derivative \( f'(x) \). Where \( c \) crosses the x-axis, \( f \) has a maximum or minimum. - **Curve \( b \) (Pink curve):** This curve has roots corresponding to the critical points of the first derivative \( f'(x) \), implying that it is the second derivative \( f''(x) \). The points where \( b \) crosses the x-axis indicate where \( f \) has inflections. - **Curve \( a \) (Green curve):** Given the changes in the concavity of \( b \), \( a \) can be deduced as the third derivative \( f'''(x) \). The roots of \( a \) indicate inflection points of the second derivative \( f''(x) \). **Explanation:** - The original function \( f(x) \) (black curve) will have inflection points where the second derivative \( f''(x) \) (pink curve) crosses the x-axis. - The first derivative \( f'(x) \) (blue curve) will intersect the x-axis at points where \( f(x) \) has local maxima and minima. - The second derivative \( f''(x) \) (pink curve) will intersect the x-axis at the inflection points of \( f(x) \), indicating changes in concavity. - The third derivative \( f'''(x)
Transcribed Image Text:### Problem 50 **Question:** The figure shows graphs of \( f \), \( f' \), \( f'' \), and \( f''' \). Identify each curve, and explain your choices. **Solution:** The given figure displays four different graphs labeled as \( a \), \( b \), \( c \), and \( d \), and we need to identify which one corresponds to the function \( f \), its first derivative \( f' \), its second derivative \( f'' \), and its third derivative \( f''' \). - **Curve \( d \) (Black curve):** This curve has inflection points and shows changes in concavity, indicating it is the original function \( f(x) \). - **Curve \( c \) (Blue curve):** This curve appears to have roots where the original function \( f(x) \) changes direction. These roots correspond to the critical points of \( f(x) \), suggesting that \( c \) is the first derivative \( f'(x) \). Where \( c \) crosses the x-axis, \( f \) has a maximum or minimum. - **Curve \( b \) (Pink curve):** This curve has roots corresponding to the critical points of the first derivative \( f'(x) \), implying that it is the second derivative \( f''(x) \). The points where \( b \) crosses the x-axis indicate where \( f \) has inflections. - **Curve \( a \) (Green curve):** Given the changes in the concavity of \( b \), \( a \) can be deduced as the third derivative \( f'''(x) \). The roots of \( a \) indicate inflection points of the second derivative \( f''(x) \). **Explanation:** - The original function \( f(x) \) (black curve) will have inflection points where the second derivative \( f''(x) \) (pink curve) crosses the x-axis. - The first derivative \( f'(x) \) (blue curve) will intersect the x-axis at points where \( f(x) \) has local maxima and minima. - The second derivative \( f''(x) \) (pink curve) will intersect the x-axis at the inflection points of \( f(x) \), indicating changes in concavity. - The third derivative \( f'''(x)
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