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.