Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=5+2 e^{-0.2 \mathrm{x}} $$. Find the intervals where $$ \mathrm{f}(\mathrm{x}) $$ 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 $$ \square $$. C. The function is decreasing on $$ \square $$. It is never increasing. Aind the location of any local extrema of $$ \mathrm{f}(\mathrm{x}) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. (Type integers or decimals. Use a comma to separate answers as needed.) B. There is a local minimum at $$ \mathrm{x}=\square $$. 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 $$ \mathrm{x}=\square $$. 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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UpStudy AI · Accepted Answer

1. Compute the derivative:

$$
\mathrm{f}(x) = 5 + 2e^{-0.2x}
$$

Differentiate using the chain rule:

$$
\mathrm{f}'(x) = 2 \cdot (-0.2)e^{-0.2x} = -0.4e^{-0.2x}
$$

Since $$ e^{-0.2x} > 0 $$ for all $$ x $$, we have

$$
\mathrm{f}'(x) < 0 \quad \text{for all } x.
$$

2. Determine intervals of increase/decrease:

Because the derivative is always negative, the function is strictly decreasing on its entire domain, $$ (-\infty, \infty) $$.

3. Identify the local extrema:

Since $$ \mathrm{f}(x) $$ is strictly decreasing, there is no point at which the derivative changes sign. Therefore, the function has no local maximum or local minimum.

4. Graph summary:

- **Y-intercept:** When $$ x=0 $$,

$$
  \mathrm{f}(0) = 5 + 2e^{0} = 5 + 2 = 7.
  $$

- **Horizontal asymptote:** As $$ x \to \infty $$, $$ e^{-0.2x} \to 0 $$ so

$$
  \mathrm{f}(x) \to 5.
  $$

- **End behavior:** As $$ x \to -\infty $$, $$ e^{-0.2x} \to \infty $$ and hence

$$
  \mathrm{f}(x) \to \infty.
  $$

- **Sketch features:** The graph starts very high for very negative $$ x $$ (heading to $$ +\infty $$) and decreases continuously to approach $$ y = 5 $$ as $$ x \to \infty $$. The graph is smooth and has a y-intercept at $$ (0,7) $$.

5. Answers based on the choices:

- Increasing/Decreasing:  
  The function is decreasing on $$(-\infty, \infty)$$ and never increasing.

- Local extrema:  
  There are no local extrema.

Therefore, the correct choices are:

For increasing/decreasing:  
**C. The function is decreasing on $$(-\infty, \infty)$$. It is never increasing.**

For local extrema:  
**D. There are no local extrema.**
The function $$ \mathrm{f}(x) = 5 + 2e^{-0.2x} $$ is strictly decreasing on $$ (-\infty, \infty) $$ and has no local extrema.

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