The function $f(x) = |x^3 + x^2 - x - 1|$ involves the absolute value of a cubic polynomial. To analyze its differentiability, we need to focus on two aspects:

1. Differentiability of the underlying cubic polynomial: The polynomial $g(x) = x^3 + x^2 - x - 1$ is differentiable everywhere because polynomials are smooth and differentiable on $\mathbb{R}$.

2. Effect of the absolute value on differentiability: The absolute value function can cause non-differentiability where the argument inside the absolute value changes sign (i.e., where $g(x) = 0$).

Steps to Analyze Differentiability:

Step 1: Find where the polynomial $g(x) = x^3 + x^2 - x - 1$ is zero

We need to solve: $x^3 + x^2 - x - 1 = 0$ By trial or synthetic division, you can find that $x = -1$ is a root. So we factor the cubic as: $x^3 + x^2 - x - 1 = (x + 1)(x^2 - 1)$ Further factoring gives: $(x + 1)(x - 1)(x + 1) = (x + 1)^2(x - 1)$ Thus, the zeros of the cubic are $x = -1$ (a double root) and $x = 1$ (a single root).

Step 2: Analyze differentiability at $x = -1$ and $x = 1$

• At $x = -1$: Since $g(x) = 0$ and $g'(x)$ is also 0 here (due to the double root), there is no cusp or corner at this point. Hence, $f(x)$ should be differentiable at $x = -1$.
• At $x = 1$: $g(x) = 0$, but $g'(x) \neq 0$. This suggests that there will be a cusp or corner at $x = 1$, making $f(x)$ non-differentiable at this point.

Step 3: Verify using the derivative

Let’s compute the derivative of $f(x)$ piecewise:

• For $g(x) \geq 0$ (i.e., $x \leq -1$ or $x \geq 1$), $f(x) = g(x)$, so $f'(x) = g'(x)$.
• For $g(x) < 0$ (i.e., $-1 < x < 1$), $f(x) = -g(x)$, so $f'(x) = -g'(x)$.

At $x = 1$, the derivative will not match from the left and right, confirming a cusp.

Step 4: Graphical Justification

Graphing $f(x)$ on a tool like Desmos would show a smooth transition through $x = -1$, but a sharp corner (non-differentiable point) at $x = 1$.

Conclusion:

• $f(x)$ is differentiable everywhere except at $x = 1$, where it has a cusp.

Would you like more details or have any questions?

Here are five follow-up questions to deepen understanding:

1. Why is differentiability lost at cusps or corners?
2. How can the double root at $x = -1$ affect smoothness and differentiability?
3. What is the graphical behavior of $g(x) = x^3 + x^2 - x - 1$ near the roots?
4. How do you interpret the role of $f'(x)$ at points of non-differentiability?
5. Can you extend this analysis to higher-degree polynomials within absolute values?

Tip: Always check for points where the argument inside the absolute value is zero when determining differentiability.