Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of $$ y=f(x) $$. $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$ 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. A. The function is concave upward on (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is concave upward on $$ \square $$. It is concave downward on $$ \square $$ (Type exact answers. Type your answers in interval notation. Use commas to separate answers as needed.) C. The function is concave downward on $$ \square $$. It is never concave upward. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) 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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Function by following steps:
- step0: Find the increasing or decreasing interval:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16}$$
- step1: Find the domain:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16},x\neq -4$$
- step2: Find the derivative:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}}$$
- step3: Find the domain:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}},x\neq -4$$
- step4: Substitute $$f^{\prime}\left(x\right)=0:$$
  $$0=-\frac{2}{\left(x+4\right)^{2}}$$
- step5: Swap the sides:
  $$-\frac{2}{\left(x+4\right)^{2}}=0$$
- step6: Rewrite the expression:
  $$\frac{-2}{\left(x+4\right)^{2}}=0$$
- step7: Cross multiply:
  $$-2=\left(x+4\right)^{2}\times 0$$
- step8: Simplify the equation:
  $$-2=0$$
- step9: The statement is false:
  $$x \in \varnothing $$
- step10: Determine the intervals:
  $$x \in \left(-\infty,-4\right)\cup \left(-4,+\infty\right)$$
- step11: Choose the points:
  $$x_{1}=-5$$
- step12: Find the values of the derivatives:
  $$f^{\prime}\left(-5\right)=-2$$
- step13: Calculate:
  $$\begin{align}&x \in \left(-\infty,-4\right)\cup \left(-4,+\infty\right)\textrm{ is decreasing interval}\end{align}$$
- step14: Evaluate:
  $$\begin{align}&\textrm{No increasing interval}\&\textrm{The decreasing interval is}\textrm{ }x \in \left(-\infty,-4\right)\cup \left(-4,+\infty\right)\end{align}$$
Analyze the inflection points of the function $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$
Function by following steps:
- step0: Find the inflection points:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16}$$
- step1: Find the domain:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16},x\neq -4$$
- step2: Find the derivative:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}}$$
- step3: Find the second derivative:
  $$f^{\prime\prime}\left(x\right)=\frac{4}{\left(x+4\right)^{3}}$$
- step4: Find the domain:
  $$f^{\prime\prime}\left(x\right)=\frac{4}{\left(x+4\right)^{3}},x\neq -4$$
- step5: Substitute $$f^{\prime\prime}\left(x\right)=0:$$
  $$0=\frac{4}{\left(x+4\right)^{3}}$$
- step6: Swap the sides:
  $$\frac{4}{\left(x+4\right)^{3}}=0$$
- step7: Cross multiply:
  $$4=\left(x+4\right)^{3}\times 0$$
- step8: Simplify the equation:
  $$4=0$$
- step9: The statement is false:
  $$x \in \varnothing $$
- step10: Find the inflection points:
  $$\textrm{No inflection points}$$
Analyze the critical points of the function $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$
Function by following steps:
- step0: Find the critical numbers:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16}$$
- step1: Find the domain:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16},x\neq -4$$
- step2: Find the derivative:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}}$$
- step3: Find the domain:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}},x\neq -4$$
- step4: Substitute $$f^{\prime}\left(x\right)=0:$$
  $$0=-\frac{2}{\left(x+4\right)^{2}}$$
- step5: Swap the sides:
  $$-\frac{2}{\left(x+4\right)^{2}}=0$$
- step6: Rewrite the expression:
  $$\frac{-2}{\left(x+4\right)^{2}}=0$$
- step7: Cross multiply:
  $$-2=\left(x+4\right)^{2}\times 0$$
- step8: Simplify the equation:
  $$-2=0$$
- step9: The statement is false:
  $$x \in \varnothing $$
- step10: Evaluate:
  $$\textrm{Critical numbers does not exist}$$
Analyze the extrema of the function $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$
Function by following steps:
- step0: Find the local extrema:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16}$$
- step1: Find the domain:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16},x\neq -4$$
- step2: Find the derivative:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}}$$
- step3: Find the domain:
  $$f^{\prime}\left(x\right)=-\frac{2}{\left(x+4\right)^{2}},x\neq -4$$
- step4: Substitute $$f^{\prime}\left(x\right)=0:$$
  $$0=-\frac{2}{\left(x+4\right)^{2}}$$
- step5: Swap the sides:
  $$-\frac{2}{\left(x+4\right)^{2}}=0$$
- step6: Rewrite the expression:
  $$\frac{-2}{\left(x+4\right)^{2}}=0$$
- step7: Cross multiply:
  $$-2=\left(x+4\right)^{2}\times 0$$
- step8: Simplify the equation:
  $$-2=0$$
- step9: The statement is false:
  $$x \in \varnothing $$
- step10: The function has no local extrema:
  $$\textrm{No local extrema}$$
Analyze the range of the function $$ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} $$
Function by following steps:
- step0: Find the range:
  $$f\left(x\right)=\frac{x^{2}+10x+24}{x^{2}+8x+16}$$
- step1: Evaluate:
  $$f\left(x\right)\neq 1$$
Let's summarize the analysis of the function $$ f(x) = \frac{x^{2}+10x+24}{x^{2}+8x+16} $$.

### 1. Inflection Points
- **Result**: There are **no inflection points**.

### 2. Intervals of Concavity
- **Result**: The function is **never concave upward** and is **concave downward** everywhere.

### 3. Critical Points and Extrema
- **Result**: There are **no critical points** and **no local extrema**.

### 4. Domain and Range
- **Domain**: The function is defined for all real numbers except where the denominator is zero. The denominator $$ x^{2}+8x+16 $$ can be factored as $$ (x+4)^{2} $$, which is zero at $$ x = -4 $$. Thus, the domain is:
  $$
  (-\infty, -4) \cup (-4, +\infty)
  $$
- **Range**: The function cannot equal 1, so:
  $$
  f(x) \neq 1
  $$

### Summary of Choices
- **A**: The function is concave upward on **(no intervals)**.
- **B**: The function is concave upward on **(no intervals)**. It is concave downward on **(-\infty, +\infty)**.
- **C**: The function is concave downward on **(-\infty, +\infty)**. It is never concave upward.

### Inflection Points
- **A**: There is an inflection point at **(no points)**.
- **B**: There are no inflection points.

### Final Answers
- **Concavity**: 
  - A: The function is concave upward on **(no intervals)**.
  - B: The function is concave downward on **(-\infty, +\infty)**.
- **Inflection Points**: 
  - B: There are no inflection points.
The function $$ f(x) = \frac{x^{2}+10x+24}{x^{2}+8x+16} $$ has no inflection points and is concave downward everywhere.

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