Math Problem Statement

Using the graph of g(x), answer the following questions: (a) Where is g(x) > 0? (b) Where is g(x) < 0? (c) Where is g(x) = 0? (d) Where is g'(x) > 0? (e) Where is g'(x) < 0? (f) Where is g'(x) = 0?

Solution

Let's analyze the graph of g(x)g(x) to answer the questions provided.

  1. Where is g(x)>0g(x) > 0?
    g(x)>0g(x) > 0 corresponds to the parts of the graph where it lies above the xx-axis.

  2. Where is g(x)<0g(x) < 0?
    g(x)<0g(x) < 0 corresponds to the parts of the graph where it lies below the xx-axis.

  3. Where is g(x)=0g(x) = 0?
    g(x)=0g(x) = 0 occurs at the xx-intercepts of the function (where the graph crosses the xx-axis).

  4. Where is g(x)>0g'(x) > 0?
    g(x)>0g'(x) > 0 indicates intervals where g(x)g(x) is increasing (where the slope is positive).

  5. Where is g(x)<0g'(x) < 0?
    g(x)<0g'(x) < 0 indicates intervals where g(x)g(x) is decreasing (where the slope is negative).

  6. Where is g(x)=0g'(x) = 0?
    g(x)=0g'(x) = 0 corresponds to the points where g(x)g(x) has local maxima, minima, or points of horizontal tangency.

I'll analyze these based on visible points and intervals in the graph and provide specific answers.Here are the answers based on analyzing the graph of g(x)g(x):

Answers

  1. (a) Where is g(x)>0g(x) > 0?

    • g(x)>0g(x) > 0 for xx in the intervals (11,6)(-11, -6) and (1,3)(1, 3).
  2. (b) Where is g(x)<0g(x) < 0?

    • g(x)<0g(x) < 0 for xx in the intervals (6,1)(-6, 1) and (3,11)(3, 11).
  3. (c) Where is g(x)=0g(x) = 0?

    • g(x)=0g(x) = 0 at x=6x = -6, x=1x = 1, and x=3x = 3.
  4. (d) Where is g(x)>0g'(x) > 0?

    • g(x)>0g'(x) > 0 for xx in the intervals (11,8)(-11, -8) and (0,2)(0, 2).
  5. (e) Where is g(x)<0g'(x) < 0?

    • g(x)<0g'(x) < 0 for xx in the intervals (8,0)(-8, 0) and (2,11)(2, 11).
  6. (f) Where is g(x)=0g'(x) = 0?

    • g(x)=0g'(x) = 0 at the local extrema, which are approximately at x=8x = -8, x=0x = 0, and x=2x = 2.

Would you like further details on any of these points?

Further Questions

  1. How can we determine the concavity of g(x)g(x) using this graph?
  2. What would the second derivative g(x)g''(x) tell us about the shape of g(x)g(x)?
  3. How can you estimate the values of g(x)g(x) at specific points if they are not labeled?
  4. What is the relationship between the intervals of g(x)>0g(x) > 0 and g(x)<0g(x) < 0 with its roots?
  5. How would adding a vertical shift to g(x)g(x) affect the intervals where g(x)g(x) is positive or negative?

Tip

When analyzing the behavior of a function, focusing on its intercepts, maxima, minima, and points of inflection provides valuable insights into its characteristics and graph shape.

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Math Problem Analysis

Mathematical Concepts

Function analysis
Graphing
Derivative
Intervals

Formulas

g(x) > 0
g(x) < 0
g(x) = 0
g'(x) > 0
g'(x) < 0
g'(x) = 0

Theorems

First derivative test for increasing/decreasing intervals
Zero-crossing points for positive/negative intervals

Suitable Grade Level

Grades 10-12