is the piecewise function f(x)=2x, x<0 and f(x)=0, x>=0 differentiable everywhere?

To determine whether the piecewise function

\[
f(x) =
\begin{cases}
2x, & \text{if } x < 0 \\
0, & \text{if } x \geq 0
\end{cases}
\]
is differentiable everywhere, we need to check two things:

1. **Continuity** at \( x = 0 \): A function must be continuous at a point for it to be differentiable there.
2. **Differentiability** at \( x = 0 \): This means checking whether the left-hand derivative and right-hand derivative are equal at \( x = 0 \).

### 1. Continuity at \( x = 0 \):
To be continuous at \( x = 0 \), we need:

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)
\]

- When \( x \to 0^- \) (approaching from the left), \( f(x) = 2x \), so:
  \[
  \lim_{x \to 0^-} f(x) = 2(0) = 0
  \]
- When \( x \to 0^+ \) (approaching from the right), \( f(x) = 0 \), so:
  \[
  \lim_{x \to 0^+} f(x) = 0
  \]
- \( f(0) = 0 \)

Since the left and right limits and the value of the function at \( x = 0 \) are all equal, the function is **continuous at \( x = 0 \)**.

### 2. Differentiability at \( x = 0 \):
Next, we check the derivative on both sides of \( x = 0 \).

- For \( x < 0 \), \( f(x) = 2x \), and the derivative is:
  \[
  f'(x) = 2
  \]
- For \( x > 0 \), \( f(x) = 0 \), and the derivative is:
  \[
  f'(x) = 0
  \]

Now, let's check the left-hand and right-hand derivatives at \( x = 0 \):

- The **left-hand derivative** as \( x \to 0^- \) is:
  \[
  \lim_{x \to 0^-} f'(x) = 2
  \]
- The **right-hand derivative** as \( x \to 0^+ \) is:
  \[
  \lim_{x \to 0^+} f'(x) = 0
  \]

Since the left-hand derivative (2) is not equal to the right-hand derivative (0), the function is **not differentiable at \( x = 0 \)**.

### Conclusion:
The function \( f(x) \) is differentiable everywhere **except at \( x = 0 \)**, where it is not differentiable due to the mismatch in the left and right derivatives.

Would you like further details or have any questions about this?

Here are five related questions you might consider:
1. How do you prove the differentiability of a piecewise function at non-zero points?
2. Can a function be continuous but not differentiable? What are common examples?
3. What are the geometric implications of differentiability at a point?
4. How does differentiability relate to smoothness of a graph?
5. What happens if we change the condition on \( f(x) \) at \( x = 0 \) to make it differentiable?

**Tip:** Continuity is a necessary condition for differentiability, but it alone does not guarantee differentiability!

Is the piecewise function f(x) = 2x, x < 0 and f(x) = 0, x >= 0 differentiable everywhere?

Explore whether the piecewise function f(x) = 2x for x < 0 and f(x) = 0 for x >= 0 is differentiable everywhere. This solution covers continuity, left-hand and right-hand derivatives, and key calculus concepts. Suitable for high school students in Grades 11-12 studying calculus or pre-calculus.

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