To solve this problem, follow the below-written steps:
Step 1: Define the function
The first step is defining the function, which is f(x) = 12+x/(2x).
Step 2: Calculate the derivative of the function
The derivative of the given function is calculated by finding the slope of the function at any point x.
Step 3: Find the critical points
Critical points happen when the derivative of the function is equal to zero or undefined. Set the derivative equal to zero and solve for x.
Step 4: Include the endpoints of the interval [0, 4] in the list of critical points
Check the values at the endpoints, which are 0 and 4 as per the given interval, because the maximum/minimum may occur here as well. Append these to the list of critical points.
Step 5: Evaluate the function at each critical point
By substituting each point from the critical points to the function, you will get the function's values at these points.
Step 6: Determine the maximum and minimum points
Maximum and minimum points refer to the points in the interval at which the function reaches its highest and lowest values, respectively.
By comparing the function evaluations at each of the critical points and the interval's endpoints, you can find at which x-values the maximum and minimum of the function on this interval occurs. For your function the maximum and minimum both occur at x=0. Hence, the absolute maximum of f(x) on the interval [0, 4] is at x=0 and the absolute minimum of f(x) on this same interval is also at x=0.