Final answer:
The absolute minimum of the function f(x)=4/(x+2) on the interval [0,100] is found by evaluating the endpoints of the interval. The function has an absolute minimum value of 4/102 at x = 100 since there are no critical points on the interval and the endpoint values are compared.
Step-by-step explanation:
To find the absolute minimum on the interval [0,100] for the function f(x)=4/(x+2), we must first understand that since f(x) is the division of a positive number by a positive expression (as x+2 is always positive for x ≥ 0), the function is always positive. Calculating the absolute minimum of a function on a closed interval involves checking the endpoints and any critical points within the interval.
First, analyze the endpoints of the interval:
- At x = 0, f(x) = 4/(0+2) = 2.
- At x = 100, f(x) = 4/(100+2) = 4/102.
Next, we need to find the critical points by taking the derivative of the function and setting it equal to zero. The derivative of f(x) is -4/(x+2)², and setting this equal to zero yields no solutions since the denominator cannot be zero. Thus, there are no critical points on the interval.
Comparing the function values at the endpoints, f(x) at x = 100 is smaller since 4/102 < 2, hence it is the minimum value on the interval [0,100]. Consequently, the absolute minimum of f(x) on [0,100] is 4/102, which occurs at x = 100.