Why does (x+ 1)/(1 +x^2) for x ∈ [0,4] have both an absolute min and max on [0,2] and determine the absolute extrema on [0,2].

Question

Consider the function f (x) := (x+ 1)/(1 +x^2) for x ∈ [0,4].

(a)Why does f possess both an absolute maximum and an absolute minimum on the interval[0,2]?

(b)Please determine the absolute extrema of f on[0,2]

Answer 1

f(0)=1 and f(2)= 3/5 ... their is a critical point at x=sqrt(2)-1 and f(.414)=1.207. Thus there is an absolute max at (.414,1.207) and an absolute min at (2-,.6)