2. Use the graph of the function f(x) to estimate where (a) f'(x)= 0, y f(x) (b) f'(x) <0 and (c) f'(x) > 0.

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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## Function Analysis - Understanding Graphs

### Problem 2

**Use the graph of the function \( f(x) \) to estimate where:**

(a) \( f'(x) = 0 \)

(b) \( f'(x) < 0 \)

(c) \( f'(x) > 0 \)

### Graph Explanation

The graph provided depicts a function \( y = f(x) \). It resembles a polynomial function with two turning points, suggestive of a quartic function. The x-axis ranges from -2 to 2, while the y-axis corresponds to \( y = f(x) \).

- **Critical Points (a - \( f'(x) = 0 \)):** 
  These occur where the curve has a horizontal tangent line, indicating potential maxima, minima, or points of inflection. On the graph, these occur approximately at the x-values where the slope of the tangent is horizontal.

- **Decreasing Intervals (b - \( f'(x) < 0 \)):** 
  These are intervals where the function is descending, shown by a downward slope on the graph. In these intervals, the derivative of the function is negative.

- **Increasing Intervals (c - \( f'(x) > 0 \)):**
  These are intervals where the function is ascending, represented by an upward slope on the graph. Here, the derivative of the function is positive.

### Problem 3

**Given the function, \( f(x) = 2x^4 - 4x^2 + 1 \):**

a. **Find and classify the critical point(s).**

b. **Find the interval(s) where \( f(x) \) is increasing.**

This problem requires you to apply calculus techniques to determine critical points and intervals of increase for the given polynomial function.
Transcribed Image Text:## Function Analysis - Understanding Graphs ### Problem 2 **Use the graph of the function \( f(x) \) to estimate where:** (a) \( f'(x) = 0 \) (b) \( f'(x) < 0 \) (c) \( f'(x) > 0 \) ### Graph Explanation The graph provided depicts a function \( y = f(x) \). It resembles a polynomial function with two turning points, suggestive of a quartic function. The x-axis ranges from -2 to 2, while the y-axis corresponds to \( y = f(x) \). - **Critical Points (a - \( f'(x) = 0 \)):** These occur where the curve has a horizontal tangent line, indicating potential maxima, minima, or points of inflection. On the graph, these occur approximately at the x-values where the slope of the tangent is horizontal. - **Decreasing Intervals (b - \( f'(x) < 0 \)):** These are intervals where the function is descending, shown by a downward slope on the graph. In these intervals, the derivative of the function is negative. - **Increasing Intervals (c - \( f'(x) > 0 \)):** These are intervals where the function is ascending, represented by an upward slope on the graph. Here, the derivative of the function is positive. ### Problem 3 **Given the function, \( f(x) = 2x^4 - 4x^2 + 1 \):** a. **Find and classify the critical point(s).** b. **Find the interval(s) where \( f(x) \) is increasing.** This problem requires you to apply calculus techniques to determine critical points and intervals of increase for the given polynomial function.
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