Question

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).
How might the following always be true that M+N=P?

Prove the relationship to be true, or state what relationship does exist between the numbers M+N and P.

Answer 1

If the maximum of function r(x) and t(x) occur at the same point c in domain P = max(r(x)+t(x)) = M+N

In general P ≤ M+N

Explanation:

If the maximum of function r(x) and t(x) occur at the same point c in domain then M=r(c) and N=t(c) So in this case P = max(r(x)+t(x)) = M+N

In general P ≤ M+N

by definition of maximum

r(x)≤M,t(x)≤N for all x in domain

=> r(x)+t(x)≤M+N for all x in domain

=> max(r(x)+t(x)) <= M+N

=> P ≤ M+N

Thus we get in general the relationship is P ≤ M+N