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(g) Explain whether every local maximum and minimum of h occurs where h' equals zero or does not exist. (h) Explain whether h has a local maximum or minimum everywhere h' equals zero or does not exist.

(g) Explain whether every local maximum and minimum of h occurs where h' equals zero or does not exist. (h) Explain whether h has a local maximum or minimum everywhere h' equals zero or does not exist.

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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Can you do g and h please

**Graph Analysis and Problem Solving for Function \( h(x) \)** **Graph Explanation:** The provided graph represents the function \( y = h(x) \). It depicts a continuous curve with several peaks and valleys, indicating points of local maxima and minima. The graph shows the behavior of \( h(x) \) from \( x = -2 \) to \( x = 3 \). **Problems and Solutions:** 6. Use the graph of \( h \) to answer the following. (a) **Find all values of \( c \) for which \( h(c) \) is a local maximum value of \( h \).** (b) **Find all values of \( c \) for which \( h(c) \) is a local minimum value of \( h \).** (c) **Does \( h \) have an absolute maximum? If so, what is its value and where does it occur?** (d) **Does \( h \) have an absolute minimum? If so, what is its value and where does it occur?** (e) **Find all values of \( c \) for which \( h'(c) = 0 \).** (f) **Find all values of \( c \) for which \( h'(c) \) does not exist.** (g) **Explain whether every local maximum and minimum of \( h \) occurs where \( h' \) equals zero or does not exist.** (h) **Explain whether \( h \) has a local maximum or minimum everywhere \( h' \) equals zero or does not exist.** **Analysis Details:** By closely analyzing the graph: - **Local Maxima:** Identify the peaks on the graph where the slope changes from positive to negative. - **Local Minima:** Identify the valleys on the graph where the slope changes from negative to positive. - **Absolute Maximum and Minimum:** Compared all peaks and valleys to determine the highest and lowest points. - **Critical Points:** Points where the derivative \( h'(c) = 0 \) (horizontal tangent) or is undefined (sharp turns or cusps). - **Sufficiency for Extrema:** Discussing conditions for the presence of local maxima and minima where derivative is zero or undefined. These analytical steps lead to a deeper understanding of how the function \( h(x) \) behaves across its domain.
Transcribed Image Text:**Graph Analysis and Problem Solving for Function \( h(x) \)** **Graph Explanation:** The provided graph represents the function \( y = h(x) \). It depicts a continuous curve with several peaks and valleys, indicating points of local maxima and minima. The graph shows the behavior of \( h(x) \) from \( x = -2 \) to \( x = 3 \). **Problems and Solutions:** 6. Use the graph of \( h \) to answer the following. (a) **Find all values of \( c \) for which \( h(c) \) is a local maximum value of \( h \).** (b) **Find all values of \( c \) for which \( h(c) \) is a local minimum value of \( h \).** (c) **Does \( h \) have an absolute maximum? If so, what is its value and where does it occur?** (d) **Does \( h \) have an absolute minimum? If so, what is its value and where does it occur?** (e) **Find all values of \( c \) for which \( h'(c) = 0 \).** (f) **Find all values of \( c \) for which \( h'(c) \) does not exist.** (g) **Explain whether every local maximum and minimum of \( h \) occurs where \( h' \) equals zero or does not exist.** (h) **Explain whether \( h \) has a local maximum or minimum everywhere \( h' \) equals zero or does not exist.** **Analysis Details:** By closely analyzing the graph: - **Local Maxima:** Identify the peaks on the graph where the slope changes from positive to negative. - **Local Minima:** Identify the valleys on the graph where the slope changes from negative to positive. - **Absolute Maximum and Minimum:** Compared all peaks and valleys to determine the highest and lowest points. - **Critical Points:** Points where the derivative \( h'(c) = 0 \) (horizontal tangent) or is undefined (sharp turns or cusps). - **Sufficiency for Extrema:** Discussing conditions for the presence of local maxima and minima where derivative is zero or undefined. These analytical steps lead to a deeper understanding of how the function \( h(x) \) behaves across its domain.
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Step 1

We know that when the slope of the tangent at a point of a function is zero then the derivative of that function at that point is zero. 

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