To analyze the function $f(x) = x^3 + x^2 - x - 1$, we can explore several aspects, including its roots, behavior, and critical points.

1. Finding Roots:

To find the roots, we can use numerical methods or graphing. However, the Rational Root Theorem suggests testing $\pm 1$.

Testing $x = 1$: $f(1) = 1^3 + 1^2 - 1 - 1 = 0$ So, $x = 1$ is a root.

We can factor $f(x)$ using synthetic division by $x - 1$: $f(x) = (x - 1)(x^2 + 2x + 1) = (x - 1)(x + 1)^2$

The roots are $x = 1$ and $x = -1$ (with multiplicity 2).

2. Behavior:

• As $x \to \infty$, $f(x) \to \infty$.
• As $x \to -\infty$, $f(x) \to -\infty$.

3. Critical Points:

To find critical points, compute the derivative: $f'(x) = 3x^2 + 2x - 1$ Setting $f'(x) = 0$: $3x^2 + 2x - 1 = 0$ Using the quadratic formula: $x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} = \frac{-2 \pm \sqrt{16}}{6} = \frac{-2 \pm 4}{6}$ This gives $x = \frac{1}{3}$ and $x = -1$.

4. Analysis of Critical Points:

• $x = -1$ is a local maximum (since it's also a root).
• $x = \frac{1}{3}$ can be analyzed further to determine if it is a minimum or maximum using the second derivative test.

5. Second Derivative:

$f''(x) = 6x + 2$ Evaluating at $x = \frac{1}{3}$: $f''\left(\frac{1}{3}\right) = 6\left(\frac{1}{3}\right) + 2 = 2 + 2 = 4 > 0$ Thus, $x = \frac{1}{3}$ is a local minimum.

Summary:

• Roots: $x = 1$ (single), $x = -1$ (double).
• Local max at $x = -1$, local min at $x = \frac{1}{3}$.

Would you like details on any specific aspect, or do you have further questions? Here are some related questions you might consider:

1. How can I graph this polynomial function?
2. What are the approximate values of the roots if calculated numerically?
3. How do the local extrema affect the overall shape of the graph?
4. Can you explain the implications of the multiplicity of roots?
5. What would the behavior of this function be if it were transformed (e.g., shifted or scaled)?

Tip: To find the behavior of polynomials at extreme values, always check the leading term's degree and coefficient.