Math Problem Statement

Identify a function f that has the given characteristics. f(-5) = f(4) = 0; f'(-0.5) = 0, f'(x) < 0 for x < -0.5; f'(x) > 0 for x > -0.5. Sketch the function.

Solution

The problem requires identifying a function f(x)f(x) based on given characteristics and sketching the appropriate graph from the provided options.

Information given:

  • f(5)=f(4)=0f(-5) = f(4) = 0: The function has roots at x=5x = -5 and x=4x = 4.
  • f(0.5)=0f'(-0.5) = 0: The derivative is zero at x=0.5x = -0.5, indicating a critical point (likely a local maximum or minimum).
  • f(x)<0f'(x) < 0 for x<0.5x < -0.5: The function is decreasing for x<0.5x < -0.5.
  • f(x)>0f'(x) > 0 for x>0.5x > -0.5: The function is increasing for x>0.5x > -0.5.

Analysis:

  1. Roots at x=5x = -5 and x=4x = 4 indicate the function crosses the x-axis at these points.
  2. There is a local extremum at x=0.5x = -0.5 because f(0.5)=0f'(-0.5) = 0. The function decreases to this point and increases afterward, indicating a local minimum at x=0.5x = -0.5.
  3. The function should be negative between the roots x=5x = -5 and x=4x = 4, except where it has local extrema.

Conclusion:

The function will likely be a cubic or a quartic polynomial, and it must have these roots and behave accordingly. Looking at the graphs, the one that satisfies all these conditions is the graph in the top left.

Reasoning for selecting the top left graph:

  • The curve crosses the x-axis at x=5x = -5 and x=4x = 4, satisfying the root conditions.
  • There is a local minimum at x=0.5x = -0.5, as required by f(0.5)=0f'(-0.5) = 0, with the function decreasing to the left of x=0.5x = -0.5 and increasing to the right of it.
  • The graph shows the function decreasing for x<0.5x < -0.5 and increasing for x>0.5x > -0.5, aligning with the derivative information.

Would you like more details on how to sketch or analyze this type of function further? Here are some related questions you might find helpful:

  1. How do we find the derivative of a function to confirm local extrema?
  2. How do critical points help us determine the shape of a graph?
  3. What is the role of concavity in identifying the type of extremum (minimum/maximum)?
  4. How does knowing the roots of a function help in graphing it?
  5. How can symmetry in a function's graph provide additional insights?

Tip: Always use the first and second derivatives of a function to analyze its behavior, including increasing/decreasing intervals and concavity.

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Math Problem Analysis

Mathematical Concepts

Calculus
Polynomial Functions
Derivative

Formulas

f'(x) = 0 (critical points)
f'(x) > 0 (increasing intervals)
f'(x) < 0 (decreasing intervals)

Theorems

First Derivative Test
Root Finding

Suitable Grade Level

Grades 11-12 (or early college calculus)