Question

Given the function f(x)=x−1002x​,

a) Find the absolute maximum and minimum (if they exist) on the interval [−100,100).
b) This part is missing, but I'll assume you're asking about the behavior of the function outside the interval [−100,100) or some related aspect.

Answer 1

To find the absolute maximum and minimum of the given function, take the derivative and set it equal to zero. Since there are no critical points on the interval, the absolute maximum and minimum occur at the endpoints. The absolute maximum is 100 and the absolute minimum is -90.

Step-by-step explanation:

a) To find the absolute maximum and minimum of the function f(x) = x - 1002x-1 on the interval [-100, 100), we can take the derivative of the function and set it equal to zero. The critical points occur when the derivative is equal to zero or undefined. However, in this case, the derivative of the function is never equal to zero or undefined, so there are no critical points on the interval. Since the function is continuous on the interval, the absolute maximum and minimum occur at the endpoints of the interval. Plugging in the endpoints, we have f(-100) = -100 - 1002(-1/-100) = -100 + 10 = -90 and f(100) = 100 - 1002(1/100) = 100 - 10.02 = 89.98. Therefore, the absolute maximum is 100 and the absolute minimum is -90.