Final Answer:
The absolute maximum of f(x) = x/(x² + 4) on the interval [-5, 5] occurs at x = -5 with a maximum value of -5/21, and the absolute minimum occurs at x = 5 with a minimum value of 5/29.
Step-by-step explanation:
To find the critical points, we first take the derivative of f(x) with respect to x and set it equal to zero:
f'(x) = (1 - x²)/(x² + 4)^2
Setting f'(x) equal to zero gives us x = ±1, but only x = 1 is within the interval [-5, 5]. Now, we evaluate f(x) at the critical point x = 1 and the endpoints x = -5 and x = 5.
f(-5) = -5/21, f(1) = 1/5, f(5) = 5/29
Therefore, the absolute maximum occurs at x = -5 with a value of -5/21, and the absolute minimum occurs at x = 5 with a value of 5/29.
It's important to note that the critical point x = 1 is not within the interval [-5, 5], so it is not considered in this context.