The number of tangents y=e^|x| at (0, 1) 

We are given the function \( y = e^{|x|} \) and need to determine the number of tangents to this function at the point \( (0, 1) \).

### Step 1: Analyze the function
The function \( y = e^{|x|} \) is defined as:

\[
y = 
\begin{cases} 
e^x & \text{if } x \geq 0, \\
e^{-x} & \text{if } x < 0.
\end{cases}
\]

### Step 2: Compute the derivatives
For \( x > 0 \), the derivative is:

\[
\frac{dy}{dx} = e^x
\]

For \( x < 0 \), the derivative is:

\[
\frac{dy}{dx} = -e^{-x}
\]

### Step 3: Consider the point \( (0, 1) \)
At \( x = 0 \):

- From the right (i.e., \( x \to 0^+ \)), the derivative \( \frac{dy}{dx} = e^0 = 1 \).
- From the left (i.e., \( x \to 0^- \)), the derivative \( \frac{dy}{dx} = -e^0 = -1 \).

### Step 4: Determine the number of tangents at \( (0, 1) \)
At the point \( (0, 1) \), the function has two different derivatives, \( 1 \) from the right and \( -1 \) from the left. This implies that there are **two distinct tangents** to the curve \( y = e^{|x|} \) at the point \( (0, 1) \).

### Conclusion:
The number of tangents to the curve \( y = e^{|x|} \) at \( (0, 1) \) is **2**.

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Would you like to explore any further details or have any questions?

Here are some related questions you might find interesting:
1. What does the existence of two tangents at a point tell us about the curve?
2. How would you find the equation of the tangents at \( (0, 1) \)?
3. What is the geometric interpretation of the absolute value function in this context?
4. Can a function have more than two tangents at a single point?
5. How do we analyze the smoothness or differentiability of functions like \( y = e^{|x|} \) at critical points?

**Tip:** When analyzing piecewise functions, always check the behavior of the derivative from both sides at critical points to understand the number of possible tangents.

Explore how to determine the number of tangents to the function y=e^|x| at the point (0, 1). This problem involves analyzing piecewise functions, derivatives, and the properties of absolute value functions. Suitable for high school students.

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