To find the absolute maximum and minimum values of a function on a specific interval, we need to first find the critical points of the function in that interval. Critical points are the values of x where the first derivative of the function is either zero or undefined. We also need to evaluate the function at the endpoints of the interval.
The function we are given here is f(x) = x - 2arctan(x).
First, let's find the derivative of this function. The derivative of the arctan(x) function is 1/(1+x^2). So, using the chain rule, the derivative of -2arctan(x) is -2/(1+x^2). The derivative of x is 1. Hence, the derivative of f(x) is 1 - 2/(1+x^2).
Setting this equal to zero to find the critical points, we find the solutions to the equation 1 - 2/(1+x^2) = 0.
Next we check the endpoints of our interval. Substituting x = 0 into our function, we get f(0) = 0 - 2arctan(0) = 0.
When we substitute x = 4, we get f(4) = 4 - 2arctan(4).
Having done these, we evaluate the function at all these points (the critical points and the endpoints of the interval), and we look for the highest (max) and lowest (min) resulting values.
When we substitute the critical points into our function, we have to ensure they fall within our interval [0,4]. If a value falls outside this interval, it's discarded because we're only interested in the interval [0,4].
After careful analysis and comparison, you find that the absolute minimum value of f(x) on this interval is 1 - π/2 and the absolute maximum value is 4 - 2arctan(4).