Do not use technology. The figure below shows the graph of f', the derivative of f. The function f is a twice differentiable function on x E (-co, co), f'(-0.8) = 0, and f"(1.3) = 0. -3 (a) For what values of x is f increasing? (Enter your answer using interval notation.) (b) For what values of x is the graph of f concave downward? (Enter your answer using interval notation.)
Do not use technology. The figure below shows the graph of f', the derivative of f. The function f is a twice differentiable function on x E (-co, co), f'(-0.8) = 0, and f"(1.3) = 0. -3 (a) For what values of x is f increasing? (Enter your answer using interval notation.) (b) For what values of x is the graph of f concave downward? (Enter your answer using interval notation.)
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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Transcribed Image Text:### Understanding the Derivative Function Through Graph Analysis
The following content explores the graphical representation of \( f' \), the derivative of a function \( f \), which is twice differentiable over the real numbers. This exercise delves into understanding how the derivative affects the behavior of the function.
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#### Graph Description
The graph presented shows \( f' \) with the \( x \)-axis ranging from approximately \(-3\) to \(3\) and the \( y \)-axis representing the derivative values.
- **Critical Points and Inflection Points**:
- \( f''(-0.8) = 0 \)
- \( f''(1.3) = 0 \)
#### Questions for Exploration
**(a) For what values of \( x \) is \( f \) increasing?**
To determine where \( f \) is increasing, observe the intervals where the graph of \( f' \) is positive.
**(b) For what values of \( x \) is the graph of \( f \) concave downward?**
To identify where \( f \) is concave downward, find the intervals where \( f' \) is decreasing, or equivalently, where \( f''(x) < 0 \).
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### Insightful Analysis
- **Increasing Function**:
- \( f \) is increasing where \( f' > 0 \).
- **Concavity**:
- \( f \) is concave downward where the graph of \( f' \) slopes downward, indicating \( f'' < 0 \).
These concepts are fundamental in understanding and visualizing the nature of functions through their derivatives, aiding in comprehensive analyses without computational tools.
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