Probem+3+Mimi,+Ian

Ian R. Wilson 3. Find the minimum value of C, given that: xy=100, x+y=C, and x is restricted to the domain of all positive, real numbers

y=100/x, and therefore, through substitution: C=x+100/x

C=x+100/x C=x+100(x^-1)

and if we express C as a function of x then:

C(x)=x+100(x^-1) and C'(x)=1+100(-x^-2) C'(x)=1-100/x^2

since the only critical points occur at x=10,x=0, and x must be a positive real number the only possible critical point is x=10, which means in turn that y=10 (since xy+100) and therefore that the minimum value of C is 20.

We really cannot be so sure that this yields the absolute minimum value! It is possible that the behavior of the function at C is such that there is no absolute minimum (who knows, perhaps the end behavior approaches negative infinity!) Check the endpoints!