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The graph of f(x) is shown. At which of the point(s) is f '(x) > 0 and f "(x) <0? D B C List all points where both f '(x) > 0 and f "(x) <0. If there is only one point where this is true, list it. If there is more than one point, list them all and separate the answers by commas. If there is no such point, type N.

The graph of f(x) is shown. At which of the point(s) is f '(x) > 0 and f "(x) <0? D B C List all points where both f '(x) > 0 and f "(x) <0. If there is only one point where this is true, list it. If there is more than one point, list them all and separate the answers by commas. If there is no such point, type N.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title:** Analyzing Critical Points of a Function **Introduction:** The graph of \( f(x) \) is shown. At which of the points is \( f ' (x) > 0 \) and \( f '' (x) < 0 \)? **Graph Analysis:** The provided graph displays a function \( f(x) \) with five marked points labeled as A, B, C, D, and E. Each point represents a critical location where the behavior of the function changes. The axes are not explicitly labeled with scale, but each point represents distinct characteristics along the curve of \( f(x) \). **Instructions:** List all points where both \( f ' (x) > 0 \) and \( f '' (x) < 0 \). If there is only one point where this is true, list it. If there is more than one point, list them all and separate the answers by commas. If there is no such point, type "N". **Graph Explanation:** - **Point A** appears to be where the function is decreasing steeply. - **Point B** is at a local minimum. - **Point C** is on a flat region where the slope seems to be zero. - **Point D** is increasing but in a flat/linear region. - **Point E** is an increasing point where the function starts curving upwards more steeply, indicating acceleration in the increase. For \( f ' (x) > 0 \) and \( f '' (x) < 0 \): - \( f ' (x) > 0 \) means the slope of the function, or the first derivative, is positive; the function is increasing. - \( f '' (x) < 0 \) means the concavity of the function, or the second derivative, is negative; the function is concave down or curves downwards. **Solution:** Evaluate each marked point with the provided criteria. - **Point A:** \( f ' (x) < 0 \), hence not applicable. - **Point B:** \( f ' (x) \approx 0 \), hence not applicable. - **Point C:** \( f ' (x) \approx 0 \), hence not applicable. - **Point D:** \( f ' (x) > 0 \) and \( f '' (x) > 0 \), hence not applicable. -
Transcribed Image Text:**Title:** Analyzing Critical Points of a Function **Introduction:** The graph of \( f(x) \) is shown. At which of the points is \( f ' (x) > 0 \) and \( f '' (x) < 0 \)? **Graph Analysis:** The provided graph displays a function \( f(x) \) with five marked points labeled as A, B, C, D, and E. Each point represents a critical location where the behavior of the function changes. The axes are not explicitly labeled with scale, but each point represents distinct characteristics along the curve of \( f(x) \). **Instructions:** List all points where both \( f ' (x) > 0 \) and \( f '' (x) < 0 \). If there is only one point where this is true, list it. If there is more than one point, list them all and separate the answers by commas. If there is no such point, type "N". **Graph Explanation:** - **Point A** appears to be where the function is decreasing steeply. - **Point B** is at a local minimum. - **Point C** is on a flat region where the slope seems to be zero. - **Point D** is increasing but in a flat/linear region. - **Point E** is an increasing point where the function starts curving upwards more steeply, indicating acceleration in the increase. For \( f ' (x) > 0 \) and \( f '' (x) < 0 \): - \( f ' (x) > 0 \) means the slope of the function, or the first derivative, is positive; the function is increasing. - \( f '' (x) < 0 \) means the concavity of the function, or the second derivative, is negative; the function is concave down or curves downwards. **Solution:** Evaluate each marked point with the provided criteria. - **Point A:** \( f ' (x) < 0 \), hence not applicable. - **Point B:** \( f ' (x) \approx 0 \), hence not applicable. - **Point C:** \( f ' (x) \approx 0 \), hence not applicable. - **Point D:** \( f ' (x) > 0 \) and \( f '' (x) > 0 \), hence not applicable. -
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