Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of

(Type an exact answer. Use a comma to separate answers as needed.)
B. The function has no local minimum.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Aype function is concave upward on the subinterval(s) ( , ).
as needed.)
B. The function is never concave upward.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The function is concave downate answers exact answer. Use a comma on the subinterval(s)
(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers
as needed.)
B. The function is never concave downward.

Ask by Ball Matthews. in the United States

Mar 12,2025

Question

Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of

(Type an exact answer. Use a comma to separate answers as needed.)
B. The function has no local minimum.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Aype function is concave upward on the subinterval(s) ( , ).
as needed.)
B. The function is never concave upward.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The function is concave downate answers exact answer. Use a comma on the subinterval(s)
(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers
as needed.)
B. The function is never concave downward.

Ask by Ball Matthews. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = e^{0.5x} + 324e^{-0.5x} $$, we will follow these steps:

1. **Find the first derivative $$ f'(x) $$** to determine critical points and local extrema.
2. **Find the second derivative $$ f''(x) $$** to analyze concavity.
3. **Determine intervals of concavity and local extrema**.

### Step 1: Find the first derivative $$ f'(x) $$

Using the rules of differentiation, we have:
$$
f'(x) = \frac{d}{dx}(e^{0.5x}) + \frac{d}{dx}(324e^{-0.5x}) = 0.5e^{0.5x} - 162e^{-0.5x}
$$

### Step 2: Find critical points

Set $$ f'(x) = 0 $$:
$$
0.5e^{0.5x} - 162e^{-0.5x} = 0
$$
This can be rearranged to:
$$
0.5e^{0.5x} = 162e^{-0.5x}
$$
Multiplying both sides by $$ e^{0.5x} $$:
$$
0.5(e^{0.5x})^2 = 162
$$
$$
(e^{0.5x})^2 = 324
$$
Taking the square root:
$$
e^{0.5x} = 18 \quad \text{or} \quad e^{0.5x} = -18 \quad (\text{not possible since } e^{0.5x} > 0)
$$
Taking the natural logarithm:
$$
0.5x = \ln(18) \implies x = 2\ln(18)
$$

### Step 3: Find the second derivative $$ f''(x) $$

Now, we find the second derivative:
$$
f''(x) = \frac{d}{dx}(0.5e^{0.5x} - 162e^{-0.5x}) = 0.25e^{0.5x} + 81e^{-0.5x}
$$

### Step 4: Analyze concavity

Since both terms $$ 0.25e^{0.5x} $$ and $$ 81e^{-0.5x} $$ are always positive for all $$ x $$, we conclude:
$$
f''(x) > 0 \quad \text{for all } x
$$
This means that the function $$ f $$ is concave upward on the entire interval $$ (-\infty, \infty) $$.

### Summary of Findings

- The function $$ f $$ has no local minimum (since it is always increasing).
- The function $$ f $$ is concave upward on the interval $$ (-\infty, \infty) $$.
- The function $$ f $$ is never concave downward.

### Final Answers

- **Local Minimum**: B. The function $$ f $$ has no local minimum.
- **Concave Upward**: A. The function $$ f $$ is concave upward on the subinterval(s) $$ (-\infty, \infty) $$.
- **Concave Downward**: B. The function $$ f $$ is never concave downward.
- **Local Minimum**: The function $$ f $$ has no local minimum. - **Concave Upward**: The function $$ f $$ is concave upward on $$ (-\infty, \infty) $$. - **Concave Downward**: The function $$ f $$ is never concave downward.

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