I need to answer 1b 2.5. Let f be a real valued function continuous on a...
Proof Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S Theorem 65.6 (a generalization of Dini's theorem) Let...
8. If f is a bounded function defined on an interval I, then prove that sụp|f | – inf|f| S sup f - inf f I I I by using Theorem 1.5.10(d) and the triangle inequality \f(x)] – f(y)] < f(x) - f(y).
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2 1 Let f: R R be...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
I literally in a desperate situation trying to prove the converse to the theorem that every continuous real valued function on a compact metric space achieves a maximum and a minimum value. The problem has been posted below, I sincerely hope any expert could give me a hand, Thanks so so much!!! Let (X, d) be a metric space. Show that if every continuous real valued function on X achieves maximum and minimum values, then X must be compact. Let...
Problem 1 You are given the maximum and minimum of the function f(x, y, z) = x2 - y2 on the surface x2 + 2y2 + 3z2 = 1 exist. Use Lagrange multiplier method to find them. Let us recall the extreme value theorem we discussed before the spring break: Extreme Value Theorem (For Functions Of Two Variables) If f(x,y) is continuous on a closed, bounded region D in the plane, then f attains a maximum value f(x,y) and a...
this is an optimization subject. that is example 2.33 Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), (G) Explain why f has a global minimum over the set Hint: Read Example 2.33 (i) Find the global minimum point and global minimum value of f over the set C. Example 2.33. Consider the function/(x1,x2)=xf+xỈ over the set The set C is not bounded, and thus the Weierstrass theorem does not guarantee the...
Let the function f: (a, b) → R is continuous in (a, b). If sup {f(x): x ∈ (a, b)} = L> 0 and inf {f(x): x ∈ (a, b)} = M <0, then prove that there is a c ∈ (a , b) such that f (c) = 0.
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...