Determine where the absolute extrema of f(x)=4xx2+1 on the interval [−4,0] occur.


**1.**  

The absolute maximum occurs at x

=

**2.**  

The absolute minimum occurs at x

=

To determine the absolute maximum and absolute minimum values of the function \( f(x) = \frac{4x}{x^2 + 1} \) on the interval \([-4, 0]\), we need to follow these steps:

1. Find the critical points of \( f(x) \) by setting its derivative \( f'(x) \) equal to zero.
2. Evaluate \( f(x) \) at the critical points and the endpoints of the given interval.
3. Compare these values to determine where the absolute maximum and minimum occur.

### Step 1: Find the critical points

First, we find the derivative of \( f(x) \):
\[ f(x) = \frac{4x}{x^2 + 1} \]

Using the quotient rule:
\[ f'(x) = \frac{(x^2 + 1)(4) - 4x(2x)}{(x^2 + 1)^2} = \frac{4x^2 + 4 - 8x^2}{(x^2 + 1)^2} = \frac{4 - 4x^2}{(x^2 + 1)^2} = \frac{4(1 - x^2)}{(x^2 + 1)^2} \]

Set the derivative equal to zero to find the critical points:
\[ \frac{4(1 - x^2)}{(x^2 + 1)^2} = 0 \]

The numerator must be zero for the fraction to be zero:
\[ 4(1 - x^2) = 0 \]
\[ 1 - x^2 = 0 \]
\[ x^2 = 1 \]
\[ x = \pm 1 \]

Since we are only considering the interval \([-4, 0]\), we take \( x = -1 \).

### Step 2: Evaluate \( f(x) \) at the critical points and endpoints of the interval

Endpoints: \( x = -4 \) and \( x = 0 \)
Critical point within the interval: \( x = -1 \)

Calculate \( f(x) \) at these points:
\[ f(-4) = \frac{4(-4)}{(-4)^2 + 1} = \frac{-16}{16 + 1} = \frac{-16}{17} \]
\[ f(0) = \frac{4(0)}{0^2 + 1} = 0 \]
\[ f(-1) = \frac{4(-1)}{(-1)^2 + 1} = \frac{-4}{1 + 1} = \frac{-4}{2} = -2 \]

### Step 3: Compare the values

Compare the values of \( f(x) \) at \( x = -4 \), \( x = -1 \), and \( x = 0 \):
\[ f(-4) = \frac{-16}{17} \approx -0.941 \]
\[ f(0) = 0 \]
\[ f(-1) = -2 \]

The maximum value is \( 0 \) at \( x = 0 \) and the minimum value is \( -2 \) at \( x = -1 \).

### Summary:

**1.** The absolute maximum occurs at \( x = 0 \).
**2.** The absolute minimum occurs at \( x = -1 \).

Would you like more details or have any questions?

### Suggested Questions:
1. How do you apply the quotient rule for differentiation?
2. What steps are involved in finding critical points of a function?
3. Why is it necessary to compare function values at critical points and endpoints?
4. Can you explain how to simplify the derivative of a rational function?
5. What does it mean for a function to have an absolute extremum?

### Tip:
When determining the extrema of a function on a closed interval, always evaluate the function at the critical points and endpoints to ensure all potential maximum and minimum values are considered.

Discover how to find the absolute maximum and minimum values of the function f(x) = 4x / (x^2 + 1) on the interval [-4, 0]. This problem involves calculus concepts such as derivatives and the application of the Extreme Value Theorem. Suitable for advanced high school students and calculus enthusiasts.

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