Consider the following statements:
I. The function `f(x)=|x|` is not differentiable at x = 1
II. The function `f(x)=e^(x)` is differentiable at x = 0
Which of the above statements is/are correct?

Options

A. Only I
B. Only II
C. Both I nor II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements regarding the differentiability of the given functions.

### Step 1: Evaluate the first statement
**Statement I:** The function \( f(x) = |x| \) is not differentiable at \( x = 1 \).

To determine if \( f(x) = |x| \) is differentiable at \( x = 1 \), we need to check the definition of differentiability. A function is differentiable at a point if the left-hand derivative and the right-hand derivative at that point are equal.

- The left-hand derivative at \( x = 1 \):
\[ f'(1^-) = \lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^-} \frac{|1 + h| - |1|}{h} = \lim_{h \to 0^-} \frac{1 + h - 1}{h} = \lim_{h \to 0^-} \frac{h}{h} = 1 \]

- The right-hand derivative at \( x = 1 \):
\[ f'(1^+) = \lim_{h \to 0^+} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^+} \frac{|1 + h| - |1|}{h} = \lim_{h \to 0^+} \frac{1 + h - 1}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 \]

Since both derivatives are equal, \( f(x) = |x| \) is differentiable at \( x = 1 \). Therefore, **Statement I is incorrect.**

### Step 2: Evaluate the second statement
**Statement II:** The function \( f(x) = e^x \) is differentiable at \( x = 0 \).

The function \( e^x \) is known to be differentiable everywhere, including at \( x = 0 \). To confirm this, we can compute the derivative:

- The derivative of \( f(x) = e^x \) is:
\[ f'(x) = e^x \]

At \( x = 0 \):
\[ f'(0) = e^0 = 1 \]

Since the derivative exists at \( x = 0 \), **Statement II is correct.**

### Conclusion
- **Statement I** is incorrect.
- **Statement II** is correct.

Thus, the correct answer is that only the second statement is true.

### Final Answer
Only Statement II is correct.

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