Surnmarize the pertinent information oblained by applying the graphing strategy and sketch the graph of $$ f(x)=\frac{x+3}{x-3} $$. A. There is a local minimum at $$ x=\square $$. There is no local maximum. (Type an integer or a decimal. Use a comma to separate enswers as needed.) B. There is a local maximum at $$ x=\square $$ and there is a local minimum at $$ x=\square $$. (Type integers or decimals. Use a comma to separale answers as needed) C. There is a local maximum at $$ x=\square $$ There is no local minmum. (Type an inleger or a decimal Use a comma to scparate answers as needed.) D. There are no local extrema

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UpStudy AI · Accepted Answer

We begin by examining the function

$$
f(x)=\frac{x+3}{x-3}.
$$

**Step 1. Domain and Vertical Asymptote**

The function is undefined when the denominator is zero:

$$
x-3=0 \quad \Rightarrow \quad x=3.
$$

Thus, there is a vertical asymptote at $$ x=3 $$.

**Step 2. Horizontal Asymptote**

Since the degrees of the numerator and denominator are identical (both are 1), the horizontal asymptote is obtained by taking the ratio of the leading coefficients:

$$
y=\frac{1}{1}=1.
$$

Therefore, the horizontal asymptote is $$ y=1 $$.

**Step 3. Intercepts**

- **x-intercept:** Set the numerator equal to zero, provided the denominator isn’t zero:

$$
  x+3=0\quad \Rightarrow \quad x=-3.
  $$

So, the graph crosses the $$ x $$-axis at $$ (-3,0) $$.

- **y-intercept:** Substitute $$ x=0 $$:

$$
  f(0)=\frac{0+3}{0-3}=\frac{3}{-3}=-1.
  $$

Thus, the graph crosses the $$ y $$-axis at $$ (0,-1) $$.

**Step 4. Monotonicity – Derivative Analysis**

Using the quotient rule, the derivative of $$ f(x) $$ is

$$
f'(x)=\frac{(x-3)(1)-(x+3)(1)}{(x-3)^2}=\frac{x-3-x-3}{(x-3)^2}=\frac{-6}{(x-3)^2}.
$$

Since $$(x-3)^2$$ is always positive (except at $$ x=3 $$ where the function is undefined), the sign of $$ f'(x) $$ is determined by $$-6$$, which is negative everywhere in the domain of $$ f(x) $$. This means the function is strictly decreasing on each interval $$ (-\infty,3) $$ and $$ (3,\infty) $$.

**Step 5. Local Extrema**

Because $$ f(x) $$ is strictly decreasing on each interval, there are no changes in the monotonicity (no turning points). Therefore, there are no local maximums or local minimums.

**Conclusion**

The correct answer is:

D. There are no local extrema.
There are no local extrema.

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