Question

Let f be the function given by f(x)= −x³/3x²−24x. What is the absolute maximum value of f on the closed interval[−6,6]?

a) 144
b) -72
c) 36
d) -36

Answer 1

To find the absolute maximum value of the function f(x) = -x³/(3x²-24x) on the closed interval [−6,6], we can first find the critical points of the function by taking the derivative and setting it equal to zero. By comparing the values at the critical point and the endpoints of the interval, the absolute maximum value is 36.

Step-by-step explanation:

To find the absolute maximum value of the function f(x) = -x³/(3x²-24x) on the closed interval [−6,6], we can first find the critical points of the function by taking the derivative and setting it equal to zero. By using the quotient rule and simplifying, we find that the derivative is equal to -x²/(x-8)². Equating this to zero, we find that x = 0 is a critical point.

Next, we evaluate the function at the endpoints of the interval. At x = -6 and x = 6, we find that f(-6) = 36 and f(6) = -36. Comparing these values with the value at the critical point, which is f(0) = 0, we can see that the absolute maximum value is 36, which corresponds to option c) in the given choices.