Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=1+3 e^{-0.2 x} $$ Find the intervals where $$ f(x) $$ is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is increasing on $$ \square $$. It is never decreasing. B. The function is increasing on $$ \square $$. It is decreasing on C. The function is decreasing on $$ \square $$. It is never increasing. Find the location of any local extrema of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. There is a local maximum at $$ x=\square $$ and there is a local minimum at $$ x=\square . \square $$. (Type integers or decimals. Use a comma to separate answers as needed.) B. There is a local minimum at $$ x=\square $$. There is no local maximum. (Type an integer or a decimal. Use a comma to separate answers as needed.) C. There is a local maximum at $$ x=\square $$. There is no local minimum. (Type an integer or a decimal. Use a comma to separate answers as needed.) D. There are no local extrema.

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1. **Finding the derivative**

We are given
   $$
   f(x) = 1 + 3e^{-0.2x}.
   $$
   To determine the intervals of increase or decrease, we differentiate:
   $$
   f'(x) = \frac{d}{dx}\left(1 + 3e^{-0.2x}\right) = 3 \cdot \left(-0.2\right)e^{-0.2x} = -0.6e^{-0.2x}.
   $$

2. **Determine the sign of $$ f'(x) $$**

Since the exponential function $$ e^{-0.2x} $$ is always positive for all real $$ x $$,
   $$
   e^{-0.2x} > 0 \quad \text{for all } x,
   $$
   it follows that
   $$
   f'(x) = -0.6e^{-0.2x} < 0 \quad \text{for all } x.
   $$
   Thus, the function $$ f(x) $$ is strictly decreasing on the entire domain $$ (-\infty, \infty) $$.

3. **Conclusion on Increase/Decrease**

Since $$ f(x) $$ is always decreasing:
   
   **Choice C is correct for interval information:**  
   The function is decreasing on $$\boxed{(-\infty,\infty)}$$. It is never increasing.

4. **Determine local extrema**

A function that is strictly decreasing on its entire domain does not have any local maximum or minimum (internal extrema). Although the endpoints or limits may define absolute behavior, they are not considered local extrema in this context.

**Choice D is correct for local extrema:**  
   There are no local extrema.

5. **Graph Summary**

- **Y-intercept:** When $$ x = 0 $$, 
     $$
     f(0) = 1 + 3e^{0} = 1 + 3 = 4.
     $$
   - **Horizontal Asymptote:** As $$ x \to \infty $$,
     $$
     e^{-0.2x} \to 0 \quad \Longrightarrow \quad f(x) \to 1.
     $$
   - **Exponential Decay:** The graph decreases continuously from the y-intercept at $$ (0,4) $$ towards its asymptote $$ y = 1 $$ as $$ x \to \infty $$. As $$ x \to -\infty $$, $$ e^{-0.2x} $$ increases without bound, and so $$ f(x) $$ increases without bound.

**Final Answers:**

- **Intervals of Increase/Decrease:**  
  The function is increasing on $$\boxed{\text{none}}$$ and decreasing on $$\boxed{(-\infty, \infty)}$$.

- **Local Extrema:**  
  There are $$\boxed{\text{no local extrema}}$$.
The function $$ f(x) = 1 + 3e^{-0.2x} $$ is strictly decreasing on $$(-∞, ∞)$$ and has no local extrema.

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