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

Answer 1

There is no upper bound for #f#.

The lower bound is #f(0,0) = 1#

Substitute #x + y = u#.

Now, optimize

#g(x,u) = x^2 - (u-x)/x#
#= x^2 + 1 - u/x#
With the only condition being #0 <= u <= 1#.
When #x# is large, the #x^2# term dominates and the #u/x# term becomes insignificant. By increasing #x#, #f# can be made arbitrarily large.
#frac{del g}{del x} = 2x + u/x^2#
From the first equation, we see that for a given value of #u#, minimum occurs when #frac{del g}{del x} = 0#. Therefore,
#x = -root(3){u/2}#
So to find the minimum of #g#, we replace all instances of #x# with #-root(3){u/2}#.
#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 #g# is increasing for #u in [0,1]#
The mimimum corresponds to #u = 0#, and consequently, #x = 0#, and #y = 0#.
The absolute minimum is #f(0,0) = 1#
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Answer 2

To maximize and minimize ( f(x, y) = x^2 - \frac{y}{x} ) constrained to ( 0 \leq x + y \leq 1 ), we need to find critical points and boundary points. Critical points occur where the gradient of ( f(x, y) ) is zero. Boundary points are where ( 0 \leq x + y \leq 1 ). Then, evaluate ( f(x, y) ) at these points and choose the maximum and minimum values.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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