A function f(x) is defined in the interval [1,4) as follows `f(x)={{:(,log_(e)[x],1 le x lt 3),(,|log_(e)x|,3 le x lt 4):}`. Then, the curve y=f(x)

Options

A. is broken at two points
B. is broken at exactly one point
C. does not have a definite tangent at two points
D. does not have a definite tangent at more than two points

Answer

To solve the problem step by step, we need to analyze the function \( f(x) \) defined in the interval \([1, 4)\):

1. **Define the function**:
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} \log_e(x) & \text{for } 1 \leq x < 3 \\ |\log_e(x)| & \text{for } 3 \leq x < 4 \end{cases} \]

2. **Evaluate the function at critical points**:
We need to evaluate \( f(x) \) at the boundaries and critical points in the given intervals:
- At \( x = 1 \):
\[ f(1) = \log_e(1) = 0 \]
- At \( x = 2 \):
\[ f(2) = \log_e(2) \approx 0.693 \]
- At \( x = 3 \):
\[ f(3) = |\log_e(3)| = \log_e(3) \approx 1.099 \]
- At \( x = 4 \) (not included in the interval):
\[ f(4) \text{ is not defined} \]

3. **Graph the function**:
- For \( 1 \leq x < 3 \): The function is \( f(x) = \log_e(x) \), which starts at \( (1, 0) \) and approaches \( (3, \log_e(3)) \).
- For \( 3 \leq x < 4 \): The function is \( f(x) = |\log_e(x)| \), which is the same as \( \log_e(x) \) since \( \log_e(x) \) is positive in this interval. The graph continues from \( (3, \log_e(3)) \) to \( (4, \log_e(4)) \).

4. **Check for continuity**:
- At \( x = 2 \): The function transitions from \( 0 \) to \( \log_e(2) \), which is continuous.
- At \( x = 3 \): The left limit \( f(3^-) = \log_e(3) \) and the right limit \( f(3^+) = \log_e(3) \) are equal. Thus, the function is continuous at \( x = 3 \).

5. **Check for differentiability**:
- At \( x = 2 \): The function is continuous, but we need to check the derivative. The derivative of \( f(x) = \log_e(x) \) is \( \frac{1}{x} \), which is defined at \( x = 2 \).
- At \( x = 3 \): The left derivative is \( \frac{1}{3} \) and the right derivative is also \( \frac{1}{3} \). Thus, the function is differentiable at \( x = 3 \).

6. **Conclusion**:
- The function \( f(x) \) is continuous on the interval \([1, 4)\) and differentiable everywhere in this interval except at the points \( x = 2 \) and \( x = 3 \).