been having a little trouble with this one:
The sum of two positive numbers is 8. Show that the square of one plus the cube of the other is at least 44.
Any help is appreciated
Let with a and b positive. Then and .
Without a loss of generality, we may choose to square and cube , so:
We want to find the minimum of this expression for .
This gives 2 critical points, but only one is in the region . So the minimum occurs at . (You should probably show that it is in fact a minimum and not a maximum.)
Now, you need to just evaluate the expression at to determine its minimum value.