(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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Can you do g and h please

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.
Expert Solution

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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