Find any relative extrema of the function. (Round your answers to three decimal places.) $$ f(x)=\arctan (x)-\arctan (x-7) $$ relative maximum $$ \quad(x, y)=(\square) $$

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To find the relative extrema of the function

$$
f(x) = \arctan(x) - \arctan(x - 7),
$$

we need to follow these steps:

1. **Find the derivative of the function** $$ f'(x) $$.
2. **Set the derivative equal to zero** to find critical points.
3. **Determine the nature of the critical points** using the second derivative test or the first derivative test.

### Step 1: Find the derivative

The derivative of $$ f(x) $$ can be calculated using the derivative of the arctangent function:

$$
\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}.
$$

Thus, the derivative $$ f'(x) $$ is:

$$
f'(x) = \frac{1}{1 + x^2} - \frac{1}{1 + (x - 7)^2}.
$$

### Step 2: Set the derivative equal to zero

Now, we set $$ f'(x) = 0 $$:

$$
\frac{1}{1 + x^2} - \frac{1}{1 + (x - 7)^2} = 0.
$$

This implies:

$$
\frac{1}{1 + x^2} = \frac{1}{1 + (x - 7)^2}.
$$

Cross-multiplying gives:

$$
1 + (x - 7)^2 = 1 + x^2.
$$

Simplifying this equation:

$$
(x - 7)^2 = x^2.
$$

Expanding the left side:

$$
x^2 - 14x + 49 = x^2.
$$

Subtracting $$ x^2 $$ from both sides:

$$
-14x + 49 = 0.
$$

Solving for $$ x $$:

$$
14x = 49 \quad \Rightarrow \quad x = \frac{49}{14} = 3.5.
$$

### Step 3: Determine the nature of the critical point

Next, we will find the second derivative $$ f''(x) $$ to determine if this critical point is a maximum or minimum.

Calculating $$ f''(x) $$:

$$
f''(x) = -\frac{2x}{(1 + x^2)^2} + \frac{2(x - 7)}{(1 + (x - 7)^2)^2}.
$$

Now we will evaluate $$ f''(3.5) $$ to determine the nature of the critical point.

Let's calculate $$ f''(3.5) $$ and check its sign.

### Step 4: Calculate $$ f(3.5) $$

We also need to find $$ f(3.5) $$:

$$
f(3.5) = \arctan(3.5) - \arctan(3.5 - 7) = \arctan(3.5) - \arctan(-3.5).
$$

Using the property of arctangent, $$ \arctan(-x) = -\arctan(x) $$:

$$
f(3.5) = \arctan(3.5) + \arctan(3.5) = 2\arctan(3.5).
$$

Now, let's calculate $$ f''(3.5) $$ and $$ f(3.5) $$ to find the relative extrema.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$2\arctan\left(3.5\right)$$
- step1: Simplify:
  $$2.584993$$
Calculate or simplify the expression $$ -2*3.5/((1 + 3.5^2)^2) + 2*(3.5 - 7)/((1 + (3.5 - 7)^2)^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-2\times 3.5}{\left(1+3.5^{2}\right)^{2}}+\frac{2\left(3.5-7\right)}{\left(1+\left(3.5-7\right)^{2}\right)^{2}}$$
- step1: Convert the expressions:
  $$\frac{-2\times 3.5}{\left(1+\left(\frac{7}{2}\right)^{2}\right)^{2}}+\frac{2\left(3.5-7\right)}{\left(1+\left(3.5-7\right)^{2}\right)^{2}}$$
- step2: Add the numbers:
  $$\frac{-2\times 3.5}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(3.5-7\right)}{\left(1+\left(3.5-7\right)^{2}\right)^{2}}$$
- step3: Subtract the numbers:
  $$\frac{-2\times 3.5}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(-3.5\right)}{\left(1+\left(3.5-7\right)^{2}\right)^{2}}$$
- step4: Subtract the numbers:
  $$\frac{-2\times 3.5}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(-3.5\right)}{\left(1+\left(-3.5\right)^{2}\right)^{2}}$$
- step5: Convert the expressions:
  $$\frac{-2\times 3.5}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(-3.5\right)}{\left(1+\left(-\frac{7}{2}\right)^{2}\right)^{2}}$$
- step6: Add the numbers:
  $$\frac{-2\times 3.5}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(-3.5\right)}{\left(\frac{53}{4}\right)^{2}}$$
- step7: Multiply the numbers:
  $$\frac{-7}{\left(\frac{53}{4}\right)^{2}}+\frac{2\left(-3.5\right)}{\left(\frac{53}{4}\right)^{2}}$$
- step8: Multiply the numbers:
  $$\frac{-7}{\left(\frac{53}{4}\right)^{2}}+\frac{-7}{\left(\frac{53}{4}\right)^{2}}$$
- step9: Divide the terms:
  $$-\frac{112}{53^{2}}+\frac{-7}{\left(\frac{53}{4}\right)^{2}}$$
- step10: Divide the terms:
  $$-\frac{112}{53^{2}}-\frac{112}{53^{2}}$$
- step11: Evaluate the power:
  $$-\frac{112}{2809}-\frac{112}{53^{2}}$$
- step12: Evaluate the power:
  $$-\frac{112}{2809}-\frac{112}{2809}$$
- step13: Transform the expression:
  $$\frac{-112-112}{2809}$$
- step14: Subtract the numbers:
  $$\frac{-224}{2809}$$
- step15: Rewrite the fraction:
  $$-\frac{224}{2809}$$
The calculations yield the following results:

1. The second derivative at $$ x = 3.5 $$ is approximately:

$$
f''(3.5) \approx -0.079744.
$$

Since $$ f''(3.5) < 0 $$, this indicates that the function has a relative maximum at this point.

2. The value of the function at $$ x = 3.5 $$ is:

$$
f(3.5) \approx 2.585.
$$

### Conclusion

The relative maximum of the function occurs at:

$$
(x, y) = (3.5, 2.585).
$$

Rounding to three decimal places, the final answer is:

$$
\text{relative maximum } (x, y) = (3.500, 2.585).
$$
The relative maximum of the function $$ f(x) = \arctan(x) - \arctan(x - 7) $$ is at $$ (3.500, 2.585) $$.

Find any relative extrema of the function. (Round your answers to three decimal places.) \[ f(x)=\arctan (x)-\arctan (x-7) \] relative maximum \( \quad(x, y)=(\square) \)

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