Math Problem Statement

Sketch a function that is continuous on left parenthesis negative infinity comma infinity right parenthesis and has the following properties. Use a number line to summarize information about the function. f prime left parenthesis x right parenthesisless than0 on left parenthesis negative infinity comma negative 2 right parenthesis​; f prime left parenthesis x right parenthesisgreater than0 on left parenthesis negative 2 comma 3 right parenthesis​; f prime left parenthesis x right parenthesisless than0 on left parenthesis 3 comma infinity right parenthesis. Question content area bottom Part 1 Which of the following graphs matches the description of the given​ properties? A. -10 10 -20 20 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve falls, crossing the x-axis at negative 2, to a minimum, and rises, crossing the x-axis at 3. B. -10 10 -20 20 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve rises, crossing the x-axis at negative 2, to a maximum, and falls, crossing the x-axis at 3. C. -10 10 -20 20 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 5 and vertical y-axis from negative 20 to 20 in increments of 5. From left to right, a smooth curve falls to a minimum at x = negative 2, rises to a maximum at x = 3, and falls. D. -10 10 -20 20 x y

Solution

To determine which graph matches the given properties, let’s analyze the behavior of f′(x)f'(x) in each interval and how that affects the shape of f(x)f(x).

Given Conditions:

  1. f′(x)<0f'(x) < 0 on (−∞,−2)(-\infty, -2): f(x)f(x) is decreasing in this interval.
  2. f′(x)>0f'(x) > 0 on (−2,3)(-2, 3): f(x)f(x) is increasing in this interval.
  3. f′(x)<0f'(x) < 0 on (3,∞)(3, \infty): f(x)f(x) is decreasing in this interval.

Implications:

  • At x=−2x = -2: The function switches from decreasing to increasing, indicating a local minimum.
  • At x=3x = 3: The function switches from increasing to decreasing, indicating a local maximum.

Analyzing Each Option:

  • Option A: The function decreases, reaches a minimum, and then increases through x=−2x = -2, and continues to increase through x=3x = 3. This does not match the conditions because it lacks a maximum at x=3x = 3.
  • Option B: The function increases, reaches a maximum at x=−2x = -2, and then decreases through x=3x = 3. This does not match the conditions because it has a maximum at x=−2x = -2 rather than a minimum.
  • Option C: The function decreases to a minimum at x=−2x = -2, increases to a maximum at x=3x = 3, and then decreases again. This matches all given conditions.
  • Option D: The graph does not match the conditions as it doesn’t clearly show the correct changes in direction.

Conclusion:

The correct answer is Option C. It aligns with the requirement that f(x)f(x) has a local minimum at x=−2x = -2, a local maximum at x=3x = 3, and the correct intervals of increase and decrease.

Would you like a more detailed explanation of why Option C is correct? Here are five related questions to expand on this concept:

  1. How does the sign of f′(x)f'(x) affect the shape of f(x)f(x) in general?
  2. What would happen if f′(x)f'(x) was positive on (−∞,−2)(-\infty, -2)?
  3. How would the graph change if f′(x)f'(x) was zero at x=−2x = -2 and x=3x = 3?
  4. Can a function have both a maximum and minimum at the same point?
  5. How would this analysis change if we were given information about f′′(x)f''(x)?

Tip: To confirm the nature of extrema, use the first or second derivative test at key points.

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Math Problem Analysis

Mathematical Concepts

Calculus
First Derivative Test
Graph Analysis

Formulas

f'(x) < 0 for decreasing intervals
f'(x) > 0 for increasing intervals

Theorems

First Derivative Test for Local Extrema

Suitable Grade Level

Grades 11-12

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