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 concave upward or downward. 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 concave upward on $$ \square $$. It is never concave downward. B. The function is concave upward on $$ \square $$. It is concave downward on $$ \square . ~ $$ C. The function is concave downward on $$ \square $$. It is never concave upward. Find the location of any inflection points of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is an inflection point at $$ x=\square $$. (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no inflection points.

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1. **Compute the derivatives.**

The function is 
   $$
   f(x)=1+3e^{-0.2x}.
   $$
   
   The first derivative is 
   $$
   f'(x)=\frac{d}{dx}\bigl(1+3e^{-0.2x}\bigr)=3\cdot(-0.2)e^{-0.2x}=-0.6e^{-0.2x}.
   $$
   
   The second derivative is
   $$
   f''(x)=\frac{d}{dx}\bigl(-0.6e^{-0.2x}\bigr)=-0.6(-0.2)e^{-0.2x}=0.12e^{-0.2x}.
   $$

2. **Determine concavity.**

Since the exponential function $$e^{-0.2x}$$ is always positive for all $$x$$, the second derivative 
   $$
   f''(x)=0.12e^{-0.2x}
   $$
   is positive for all real $$x$$. Therefore, the function is concave upward on 
   $$
   (-\infty, \infty).
   $$
   It is never concave downward.

3. **Determine the inflection points.**

An inflection point occurs when the concavity changes, which corresponds to when $$f''(x)=0$$ or is undefined. Here, 
   $$
   f''(x)=0.12e^{-0.2x}
   $$
   is never zero and is defined for all $$x$$. Therefore, there are no inflection points.

4. **Graphing strategy summary.**

- The function $$f(x)=1+3e^{-0.2x}$$ has a horizontal asymptote at $$y=1$$ since 
     $$
     \lim_{x\to\infty}e^{-0.2x}=0.
     $$
   - At $$x=0$$, $$f(0)=1+3=4$$.
   - The function is strictly decreasing because $$f'(x)=-0.6e^{-0.2x}<0$$ for all $$x$$.
   - The graph is concave upward on all of $$(-\infty, \infty)$$.

5. **Final selections:**

- Concavity:
     - **A.** The function is concave upward on $$\mathbf{(-\infty, \infty)}$$. It is never concave downward.
     
   - Inflection point:
     - **B.** There are no inflection points.

Thus, the correct answers are:

- **For concavity:** The function is concave upward on $$(-\infty, \infty)$$. It is never concave downward.
- **For inflection points:** There are no inflection points.
The function $$ f(x) = 1 + 3e^{-0.2x} $$ is concave upward on $$(-\infty, \infty)$$ and has no inflection points.

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