Math Homework Help Needed

[mommartz]

Suppose r(x) and t(x) are two functions with the same domain, and let h(x)=r(x)+t(x).
Suppose also that each of the 3 functions r, t and h, has a maximum value in this domain (i.e. a value that is greater than or equal to all the other values of the function).

• Let M = the maximum value of r(x),
• N = the maximum value of t(x), and
• P = the maximum value of h(x).

Must it be true that M+N=P?
Prove this, or state what relationship does exist between the numbers M+N and P.

 

[tkhunny]

Suppose r(x) and t(x) are two functions with the same domain, and let h(x)=r(x)+t(x).
Suppose also that each of the 3 functions r, t and h, has a maximum value in this domain (i.e. a value that is greater than or equal to all the other values of the function).

• Let M = the maximum value of r(x),
• N = the maximum value of t(x), and
• P = the maximum value of h(x).

Must it be true that M+N=P?
Prove this, or state what relationship does exist between the numbers M+N and P.

What if the maxima of r and t don't happen in the same place (for the same value of x)?

Have you considered r(x) = -t(x)?

Plenty to think about. If you hadn't said "or equal to", it might be little harder to dream up a counter-example.

 

[Steven G]

Suppose r(x) and t(x) are two functions with the same domain, and let h(x)=r(x)+t(x).
Suppose also that each of the 3 functions r, t and h, has a maximum value in this domain (i.e. a value that is greater than or equal to all the other values of the function).

• Let M = the maximum value of r(x),
• N = the maximum value of t(x), and
• P = the maximum value of h(x).

Must it be true that M+N=P?
Prove this, or state what relationship does exist between the numbers M+N and P.

Suppose r(x) has a max at x=a with r(a) =M and t(x) has a max at x=b (a\(\displaystyle \neq\)b) with t(b)=N.

Now for every x = c, h(c) = r(c) + t(c) <= M + N. The question is if the <= sign can be replaced with < sign?? Think about it!