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

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

Ask by Marsh Lambert. in the United States

Mar 12,2025

Question

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

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

Ask by Marsh Lambert. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

1. **Function and its derivatives**

The function is 
   $$
   f(x)=e^{0.5x}+324\,e^{-0.5x}.
   $$
   
   The first derivative is obtained by differentiating term‐by‐term:
   $$
   f'(x)=0.5\,e^{0.5x}-162\,e^{-0.5x}.
   $$
   
   The second derivative is
   $$
   f''(x)=0.25\,e^{0.5x}+81\,e^{-0.5x}.
   $$

2. **Concavity analysis**

Since the exponential function is positive for all $$x$$, both terms in
   $$
   f''(x)=0.25\,e^{0.5x}+81\,e^{-0.5x}
   $$
   are positive. Hence,
   $$
   f''(x)>0\quad\text{for all }x.
   $$
   This implies that the function is always concave upward and is never concave downward.
   
   Therefore, from the options provided:
   
   - For concavity:  
     **B. The function $$ f $$ is never concave downward.**

3. **Inflection point analysis**

An inflection point occurs when the concavity of the function changes, i.e., when $$ f''(x) $$ changes sign. Since
   $$
   f''(x)>0\quad\text{for all }x,
   $$
   there is no change in concavity and thus no inflection point.
   
   Therefore, from the options provided:
   
   - For inflection points:  
     **B. The function $$ f $$ has no inflection point.**

4. **Additional information from graphing**

- **Behavior at infinity:**  
     As $$ x\to\infty $$, the term $$ e^{0.5x} $$ dominates, so $$ f(x) \to \infty $$.  
     As $$ x\to -\infty $$, the term $$ 324\,e^{-0.5x} $$ dominates, so $$ f(x) \to \infty $$.

- **Finding the minimum:**  
     To locate the minimum, set the first derivative equal to zero:
     $$
     0.5\,e^{0.5x}-162\,e^{-0.5x}=0.
     $$
     Solving for $$ x $$:
     $$
     \begin{aligned}
     0.5\,e^{0.5x} &= 162\,e^{-0.5x} \
     e^{0.5x}\cdot e^{0.5x} &= 324 \quad (\text{multiplying both sides by }e^{0.5x}) \
     e^x &= 324 \
     x &= \ln(324).
     \end{aligned}
     $$
     Substitute $$ x=\ln(324) $$ into $$ f(x) $$:
     $$
     \begin{aligned}
     f(\ln(324)) &= e^{0.5\ln(324)}+324\,e^{-0.5\ln(324)} \
     &= 324^{0.5}+324\,(324^{-0.5}) \
     &= 18+18=36.
     \end{aligned}
     $$
     Hence, there is a minimum at the point $$ \bigl(\ln(324),\,36\bigr) $$.

5. **Summary of Answers**

- Concavity:  
     **B. The function $$ f $$ is never concave downward.**

- Inflection points:  
     **B. The function $$ f $$ has no inflection point.**

6. **Sketching the graph**

The graph of $$ y=f(x) $$ has the following characteristics:
   
   - It has a minimum point at $$ \left(\ln(324),\,36\right) $$.
   - As $$ x\to-\infty $$ and $$ x\to\infty $$, the function tends to infinity.
   - The function is always concave upward (convex) since $$ f''(x)>0 $$ for all $$ x $$.

The sketch should include a smooth curve that descends to the minimum point at $$ \left(\ln(324),\,36\right) $$ and then increases on both sides, remaining concave upward everywhere.
The function $$ f(x) = e^{0.5x} + 324e^{-0.5x} $$ is always concave upward and has no inflection points. It has a minimum at $$ x = \ln(324) $$ with a value of 36.

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