Let `f(x)={{:("max".{e^x,e^(-x),2},xle0),("min".{e^x,e^(-x),2},xgt0):}`
Which of the following statements is NOT true?

Options

A. f (x) is discontinuous at x = 0
B. f (x) is non-derivable at exactly two points
C. f (x) has non-removable type of discontinuity at x = 0 with jump of discontinuity equal to 2.
D. f (x) is continuous but non-derivable at `x = log_(e)""(1)/2`

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} \max(e^x, e^{-x}, 2) & \text{if } x \leq 0 \\ \min(e^x, e^{-x}, 2) & \text{if } x > 0 \end{cases} \]

### Step 1: Analyze \( f(x) \) for \( x \leq 0 \)

For \( x \leq 0 \):
- \( e^x \) is decreasing and approaches 1 as \( x \) approaches 0.
- \( e^{-x} \) is increasing and approaches 1 as \( x \) approaches 0.
- The constant function \( y = 2 \) is always above \( y = e^x \) for \( x \leq 0 \) since \( e^x \) approaches 1.

Thus, for \( x \leq 0 \):
- \( f(x) = \max(e^x, e^{-x}, 2) = 2 \) since \( 2 > e^x \) and \( 2 > e^{-x} \).

### Step 2: Analyze \( f(x) \) for \( x > 0 \)

For \( x > 0 \):
- \( e^x \) is increasing and approaches infinity as \( x \) increases.
- \( e^{-x} \) is decreasing and approaches 0 as \( x \) increases.
- The constant function \( y = 2 \) is above \( e^{-x} \) for all \( x > 0 \) since \( e^{-x} < 2 \).

Thus, for \( x > 0 \):
- \( f(x) = \min(e^x, e^{-x}, 2) = e^{-x} \) since \( e^{-x} < 2 \) and \( e^{-x} < e^x \).

### Step 3: Evaluate \( f(0) \)

At \( x = 0 \):
- \( f(0) = \max(e^0, e^0, 2) = \max(1, 1, 2) = 2 \).

### Step 4: Check continuity at \( x = 0 \)

To check continuity at \( x = 0 \):
- \( \lim_{x \to 0^-} f(x) = 2 \)
- \( \lim_{x \to 0^+} f(x) = e^{-0} = 1 \)
- Since \( \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \), \( f(x) \) is discontinuous at \( x = 0 \).

### Step 5: Identify the statements

Now, we need to evaluate the statements given in the options:

1. \( f(x) \) is discontinuous at \( x = 0 \) - **True**
2. \( f(x) \) is not derivable at exactly 2 points - **True** (not derivable at \( x = 0 \) and at \( x = \ln(1/2) \))
3. \( f(x) \) has non-removable type discontinuity at \( x = 0 \) with a jump of discontinuity equal to 2 - **False** (the jump is actually 1)
4. \( f(x) \) is continuous but not derivable at \( x = \ln(1/2) \) - **True**

### Conclusion

The statement that is NOT true is:
- **Option 3: \( f(x) \) has non-removable type discontinuity at \( x = 0 \) with a jump of discontinuity equal to 2.**