Find the maximum of , subject to the condition
ive got the x values as: .
. . and y values: . . . . . Right . . . good work!
So and are the maximums
but how do i show that they are maximums?
There must be a test for this (even with Lagrange multipliers),
. . but I'm not familiar with it.
But it is clear that and must have the same sign
. . in order for to be a maximum.
That may be sufficient . . . or not.
please check your calculations. I've got the same x-values but the y-value of the maxima is +1:
(You only considered the condition without using the equation of the function)
To show that a certain value is indeed a maximum you have to calculate the 2nd derivative. Plug in the x-value. If f''(x) < 0 then you have a maximum.
Doing this we find that , and so the corresponding 's are and .
(note in doing this I had to assume that , so if the maximum turns out to be negative we will need to throw in the possibility that give a maximum for of )
This gives four posible points for the maximum .
Now we could do some clever things to determine which of these corresponds to the maximum, but the easiest way is just to evaluate at each point and the largest will be the maximum.