Math Problem Statement

The figure shows the graphs of f, f', and f''. Identify each curve.

Solution

To identify each curve in the graph as ff, ff', and ff'', let's analyze their shapes and characteristics:

  1. Function f(x)f(x): This is the original function, and it will have a smoother curve, possibly with turning points (maximums and minimums).
  2. Derivative f(x)f'(x): The first derivative will indicate the slope of f(x)f(x). It crosses the x-axis at the points where f(x)f(x) has maxima and minima (since f=0f' = 0 at these points).
  3. Second Derivative f(x)f''(x): The second derivative represents the concavity of f(x)f(x). It crosses the x-axis at points where f(x)f(x) changes concavity (inflection points).

Analysis of the Graphs

  • Curve aa (Purple): This curve is smooth and has clear turning points, making it likely to be the original function f(x)f(x).
  • Curve bb (Yellow): This curve crosses the x-axis at the turning points of aa, indicating it represents f(x)f'(x), as it shows the slope changes of ff.
  • Curve cc (Cyan): This curve crosses the x-axis where bb has its turning points, indicating inflection points in ff, so it likely represents f(x)f''(x).

Conclusion

  • f(x)f(x) is represented by Curve aa (Purple).
  • f(x)f'(x) is represented by Curve bb (Yellow).
  • f(x)f''(x) is represented by Curve cc (Cyan).

Let me know if you need further explanation!

Related Questions:

  1. What are inflection points, and how do they relate to the second derivative?
  2. How can we tell if f(x)f(x) has a local maximum or minimum using f(x)f'(x)?
  3. How does concavity relate to the sign of f(x)f''(x)?
  4. What does it mean when f(x)=0f''(x) = 0?
  5. How would you sketch ff' if you were given only ff?

Tip:

To analyze graphs involving derivatives, look at where curves cross the x-axis, as these points often indicate critical behaviors (like maxima, minima, and inflection points) of the original function.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Inflection Points
Critical Points
Graph Analysis

Formulas

f'(x) represents the slope of f(x)
f''(x) indicates concavity changes in f(x)

Theorems

First Derivative Test
Second Derivative Test

Suitable Grade Level

High School (Grades 11-12) or Early College