Question

The maximum value of the objective function \((x+y)\) subject to the constraint \((x^2+y^2)=2\) is (in integer).

Hint:

For constrained optimization involving \(x^2+y^2\), identities like \((x+y)^2=x^2+y^2+2xy\) and inequalities such as AM-GM are very useful.

Answer 1

Correct Answer: 2

Step 1: Identify the objective function and constraint.
The objective function is
\[ Z=x+y \] subject to the constraint
\[ x^2+y^2=2 \]
We need to find the maximum possible value of
\[ x+y \] under the given constraint.

Step 2: Use the identity involving squares.
Consider the square of the objective function:
\[ (x+y)^2=x^2+y^2+2xy \]
Using the constraint
\[ x^2+y^2=2 \] we get
\[ (x+y)^2=2+2xy \]

Step 3: Apply the AM-GM inequality.
We know that
\[ x^2+y^2 \geq 2xy \]
Since
\[ x^2+y^2=2 \] therefore,
\[ 2 \geq 2xy \] \[ xy \leq 1 \]
Substituting the maximum possible value of \(xy\) into the expression,
\[ (x+y)^2=2+2(1) \] \[ (x+y)^2=4 \]
Hence,
\[ x+y \leq 2 \]

Step 4: Check when equality occurs.
Equality in
\[ x^2+y^2 \geq 2xy \] occurs when
\[ x=y \]
Using the constraint,
\[ x^2+x^2=2 \] \[ 2x^2=2 \] \[ x^2=1 \] \[ x=1 \] and
\[ y=1 \]
Thus,
\[ x+y=1+1=2 \]

Step 5: Verify the maximum value.
The point
\[ (1,1) \] satisfies the constraint
\[ 1^2+1^2=2 \]
and gives the objective value
\[ 2 \]
Therefore, this is indeed the maximum value.

Step 6: Final conclusion.
Hence, the maximum value of
\[ x+y \] subject to
\[ x^2+y^2=2 \] is
\[ \boxed{2} \]