Properties of Integrals

[jdz86]

Homework Statement

A.) Prove: $$m_{i}$$(f) + $$m_{i}$$(g) $$\leq$$ $$m_{i}$$(f+g) $$\leq$$ $$M_{i}$$(f+g) $$\leq$$ $$M_{i}$$(f) + $$M_{i}$$(g)

B.) Prove: $$M_{i}$$($$\alpha$$f) = {$$\alpha$$$$M_{i}$$(f) $$\alpha$$ $$\succ$$ 0 or $$\alpha$$$$m_{i}$$(f) $$\alpha$$ $$\prec$$ 0
and
$$m_{i}$$($$\alpha$$f) = {$$\alpha$$$$m_{i}$$(f) $$\alpha$$ $$\succ$$ 0 or $$\alpha$$$$M_{i}$$(f) $$\alpha$$ $$\prec$$ 0

C.) Take P = $$P_{1}$$ = {a,b}, and write $$m_{i}$$ = m, $$M_{i}$$ = M (i=1). Give an example for each of the following:
(a) m(f) + m (g) $$\prec$$ m(f+g)
(b) M(f+g) $$\prec$$ M(f) + M(g)
(c) both (a) and (b) hold

Homework Equations

P = {$$x_{0}$$,...,$$x_{n}$$} and is a partition of [a,b]
$$M_{i}$$ = {sup {f(x) : x $$\in$$ [$$x_{i-1}$$, $$x_{i}$$]}}
$$m_{i}$$ = {inf {f(x) : x $$\in$$ [$$x_{i-1}$$, $$x_{i}$$]}}
$$\alpha$$ is just any multiplier

The Attempt at a Solution

For A.) and B.) i was going to take different sets and show it that way, but it will be a long drawn out process, i can't put my finger on the "shortcut/abbreviated" version. Any suggestions on what method to use??

And for C.) (a) f = [-2,-1) g = [-1,0]
(b) f = [0,1) g = [1,2]
(c) ??

 

[mighty2000]

I'm not sure about this one, I think it would depend on the specific choice of f and g. Can you give me a hint on how to approach this one?

Hi there! Great question. There are a few different ways you could approach these proofs, so I'll provide a couple of different options for each part.

A.) One approach would be to use the definitions of m_i and M_i in terms of sups and infs, and then use properties of sups and infs to show that the inequalities hold. Another approach would be to use the fact that the sum and product of two functions are continuous and then use the Intermediate Value Theorem to show that the inequalities hold.

B.) For the first part, you could use the definition of M_i and m_i in terms of sups and infs, and then use properties of sups and infs to show that the equality holds. For the second part, you could use the definition of \alpha \succ 0 and \alpha \prec 0 to show that the inequalities hold.

C.) For (a), you could choose a function f that has a large jump or discontinuity at x = -1 and another function g that has a large jump or discontinuity at x = 0, and then show that the left side is smaller than the right side. For (b), you could choose functions f and g that have small jumps or discontinuities at x = 1 and x = 2, respectively, and then show that the left side is smaller than the right side. For (c), you could choose functions f and g that have large jumps or discontinuities at both x = -1 and x = 1, and then show that both inequalities hold.

I hope this helps! Let me know if you have any other questions or if you would like me to go into more detail on any of these approaches.