Notice that \\The Closed Interval Method\" to find absolute maximum requires the

Question

Notice that The Closed Interval Method" to find absolute maximum requires the function f to be
continuous on the closed interval. Give one example of f to each of the following cases (in each case,
you need to give a formula for f, showing graphically will not be sufficient, but definitely reinforce
your answer):
(a) f is not continuous but still has absolute maximum and absolute minimum values over the
interval [0, 1].
(b) f is not continuous and has no absolute maximum over the interval [0, 1].

Explanation / Answer

a) consider the lagranges function f : R -> {0,1} f(x) = 1 if x is rational f(x) = 0 if x is irrational restrict the function to [0,1] g : [0,1] -> {0,1} g(x) = 1 if x is rational g(x) = 0 if x is irrational Now g is not continuous at any point in [0,1] but its bounded and has absolute maximum and minimum and they are attained. if you want another simpler example : f(x) = 10 if x = 0 = -x else this f is discontinuous at x=0 but x has absolute maxima (at x=0) and absolute minima at x = 1 b) consider this fn f(0) = 0 and f(x) = 1/x this function has no absolute maxima although it does have an absolute minima message me if you have any doubts

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.