Question 944534
Let's call the function {{{S}}}.
{{{S=x^2+y^2}}}
Since you know the relationship between {{{x}}} and {{{y}}}, you can make {{{S}}} a function of one variable.
{{{4x+y=11}}}
{{{y=-4x+11}}}
So then,
{{{S=x^2+(-4x+11)^2}}}
{{{S=x^2+(16x^2-88x+121)}}}
{{{S=17x^2-88x+121}}}
To find an extremum, take the derivative and set it equal to zero.
You could also convert to vertex form by completing the square.
{{{dS/dx=34x-88=0}}}
{{{34x=88}}}
{{{x=88/34}}}
{{{highlight(x=44/17)}}}
Then,
{{{y=-4(44/17)+11}}}
{{{y=-176/17+187/17}}}
{{{highlight(y=11/17)}}}