Math Problem Statement

Compare the maximum values and end behavior of the polynomial functions f and g, where f is given graphically, and g(x) = x^2(9 - x).

Solution

Based on the information in the uploaded image, here is the analysis and solution:

  1. Maximum Values of f(x)f(x) and g(x)g(x):

    • From the graph, the maximum value of f(x)f(x) is 100100.
    • The maximum value of g(x)=x2(9x)g(x) = x^2(9 - x) is computed as follows:
      • g(x)=x2(9x)g(x) = x^2(9 - x) is a cubic function. To find its maximum, we differentiate and solve g(x)=0g'(x) = 0: g(x)=2x(9x)x2=18x3x2=3x(6x).g'(x) = 2x(9 - x) - x^2 = 18x - 3x^2 = 3x(6 - x). The critical points are x=0x = 0 and x=6x = 6.
      • Evaluating g(x)g(x) at x=6x = 6: g(6)=62(96)=363=108.g(6) = 6^2(9 - 6) = 36 \cdot 3 = 108. Thus, g(x)g(x) has a maximum value of 108108, which is higher than the maximum value of f(x)f(x).
  2. Comparison of End Behavior:

    • For f(x)f(x), as xx \to \infty, yy \to -\infty, as this is a downward-opening polynomial.
    • For g(x)=x2(9x)g(x) = x^2(9 - x), as xx \to \infty, yy \to -\infty, because the cubic term x3-x^3 dominates.

Correct Answer for End Behavior:

  • The correct answer is C: "For both functions, when xx \to \infty, yy \to -\infty."

Let me know if you'd like a step-by-step explanation of any part!

Questions for Expansion:

  1. Would you like a detailed derivation of the derivative for g(x)g(x) to confirm its maximum point?
  2. How do the graphs of f(x)f(x) and g(x)g(x) differ in behavior at x0+x \to 0^+?
  3. What is the role of the leading term in determining the end behavior of these polynomials?
  4. Can we analyze the symmetry of g(x)g(x) in comparison to f(x)f(x)?
  5. Would you like a general discussion on finding maxima of polynomials?

Tip:

For any polynomial, the degree of the leading term is the key determinant of end behavior—always focus on this term for quick insights!

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Math Problem Analysis

Mathematical Concepts

Polynomials
Cubic Functions
Critical Points
End Behavior

Formulas

g'(x) = 2x(9 - x) - x^2 = 18x - 3x^2 = 3x(6 - x)
Maximum value of g(x) at x = 6: g(6) = 6^2(9 - 6) = 108

Theorems

Differentiation to find critical points
Polynomial end behavior determined by leading term

Suitable Grade Level

Grades 10-12