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请解答图片上的三道数学选择题

Let's go through the questions one by one.

### Question 1:
**Question:** 如果函数\( f(x) \)在区间\( (a, b) \)内恒有\( f'(x) > 0, f''(x) < 0 \)，则函数的图像在\( (a, b) \)内是（）。

**Options:**
A. 单调递增，曲线是凹的  
B. 单调递减，曲线是凹的  
C. 单调递增，曲线是凸的  
D. 单调递减，曲线是凸的  

**Solution:**
Given \( f'(x) > 0 \) means the function \( f(x) \) is increasing (monotonically increasing) on \( (a, b) \).

Given \( f''(x) < 0 \) means the function \( f(x) \) is concave down (the curve is concave).

So, the correct answer is:
**A. 单调递增，曲线是凹的**

### Question 2:
**Question:** 函数\( y = f(x) \)与其反函数\( y = f^{-1}(x) \)的图形对称于直线（）。

**Options:**
A. \( x = 0 \)  
B. \( y = 0 \)  
C. \( y = x \)  
D. \( y = -x \)  

**Solution:**
The graph of a function \( y = f(x) \) and its inverse \( y = f^{-1}(x) \) are symmetric with respect to the line \( y = x \).

So, the correct answer is:
**C. \( y = x \)**

### Question 3:
**Question:** 函数\( f(x) = \begin{cases} 
xe^x, & x < 0 \\
x, & x \ge 0 
\end{cases} \)在点\( x = 0 \)处（）。

**Options:**
A. 可导，连续  
B. 连续  
C. 可微  
D. 连续不可导  

**Solution:**
First, check for continuity at \( x = 0 \):

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

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

Since both limits are equal to \( f(0) = 0 \), the function is continuous at \( x = 0 \).

Next, check for differentiability at \( x = 0 \):

The left-hand derivative at \( x = 0 \):

\[ \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{he^h - 0}{h} = \lim_{h \to 0^-} e^h = 1 \]

The right-hand derivative at \( x = 0 \):

\[ \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h - 0}{h} = 1 \]

Since both left-hand and right-hand derivatives are equal, \( f(x) \) is differentiable at \( x = 0 \).

So, the correct answer is:
**A. 可导，连续**

Would you like more details or have any questions on any of these problems?

Here are 5 more questions for you to consider:
1. What are the necessary conditions for a function to be continuous at a point?
2. How do you determine if a function is concave up or concave down?
3. What is the geometric interpretation of the derivative of a function?
4. How do you find the inverse of a function?
5. What are the steps to check if a function is differentiable at a point?

**Tip:** Always verify continuity before checking differentiability, as differentiability implies continuity.

Explore and solve three challenging calculus problems: understanding function behavior, symmetry in function graphs, and continuity and differentiability of piecewise functions. Detailed solutions and explanations provided.

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