Math Problem Statement

f(x)=6x3−2x2−3x−3 graphf\left(x\right)=6x^3-2x^2-3x-3\text{ graph}

Solution

The graph of the function f(x)=6x3−2x2−3x−3f(x) = 6x^3 - 2x^2 - 3x - 3 shows a cubic curve with:

  • A y-intercept at (0,−3)(0, -3).
  • The curve passing through several key points, with typical cubic behavior: rising steeply to the right and falling steeply to the left.
  • Potential turning points (local maxima and minima) identified by solving the first derivative.

Would you like to see more details on solving for critical points, inflection points, or x-intercepts? Or do you have any specific questions?

Related Questions:

  1. How do we solve cubic equations for their exact roots?
  2. What are the steps to find the turning points and classify them as maxima or minima?
  3. How does the second derivative test work for finding points of inflection?
  4. Can we approximate the x-intercepts numerically for this function?
  5. What is the significance of end behavior in polynomial functions?

Tip: Understanding how to derive and interpret critical points helps in sketching accurate graphs and analyzing functions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Functions
Derivatives
Turning Points
End Behavior

Formulas

Cubic function: f(x) = ax^3 + bx^2 + cx + d
First derivative: f'(x) for critical points
Second derivative: f''(x) for inflection points

Theorems

Fundamental Theorem of Algebra
First Derivative Test for Local Extrema
Second Derivative Test for Inflection Points

Suitable Grade Level

Grades 10-12

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