The image shows a question related to differentiability. Let me summarize and analyze it:

The question seems to be asking to show that the function: $f(x) = |x| + |x - 1|$ is not differentiable at $x = 1$.

To determine if $f(x)$ is differentiable at $x = 1$, we can check the behavior of the derivative from the left and right of this point. The function $f(x) = |x| + |x - 1|$ involves absolute values, which typically leads to points where differentiability may fail, especially at points where the argument of the absolute value changes sign (here, $x = 0$ and $x = 1$).

Steps to Determine Differentiability at $x = 1$

1. Consider the behavior for $x < 1$:

   • For $x < 1$, $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.
   • $|x - 1| = 1 - x$.

   This gives two cases to consider within $x < 1$:

   • If $0 \le x < 1$: $f(x) = x + (1 - x) = 1$.
   • If $x < 0$: $f(x) = -x + (1 - x) = 1 - 2x$.

2. For $x = 1$:

   • We can calculate the left-hand and right-hand limits of the derivative at $x = 1$.

3. Calculate the derivatives on either side of $x = 1$:

   • For $x > 1$, $f(x) = x + (x - 1) = 2x - 1$, which has a derivative of $2$.
   • For $x < 1$ but $x \ge 0$, $f(x)$ is constant at $f(x) = 1$, so the derivative is $0$.

4. Conclusion:

   • The left-hand derivative at $x = 1$ is $0$, and the right-hand derivative is $2$.
   • Since the left-hand and right-hand derivatives do not match, $f(x)$ is not differentiable at $x = 1$.

Would you like a detailed step-by-step breakdown, or do you have further questions?

Related Questions:

1. How do we evaluate the differentiability of a function with absolute values?
2. What are other methods to check differentiability at a given point?
3. Why does the presence of an absolute value function often lead to non-differentiability?
4. Can you find points of non-differentiability for $f(x) = |x| + |x - 1|$ other than $x = 1$?
5. How does the behavior of the derivative near $x = 1$ help in determining differentiability?

Tip:

For functions involving absolute values, consider breaking down the function into cases based on the intervals where the expressions inside the absolute values are positive or negative.