# How do you maximize and minimize #f(x,y)=x^2-y/x# constrained to #0<=x+y<=1#?

##### 1 Answer

There is no upper bound for

The lower bound is

#### Explanation:

Substitute

Now, optimize

#g(x,u) = x^2 - (u-x)/x#

#= x^2 + 1 - u/x#

With the only condition being

When

#frac{del g}{del x} = 2x + u/x^2#

From the first equation, we see that for a given value of

#x = -root(3){u/2}#

So to find the minimum of

#g(u) = (-root(3){u/2})^2 + 1 - u/(-root(3){u/2})#

# = 2^{-2/3}*u^{2/3} + 1 + root(3)2*u^{2/3}#

# = 3root(3){u^2/4} + 1 #

Even without differentiating, it is quite easy to see that

The mimimum corresponds to

The absolute minimum is