Minimum point with constraint

Question

$$f(x,y)= x^2 + y^2$$ and $$g(x,y)=x+y-10$$ How can I find the minimum point? I solved it and found that $x=y=5$ I plugged what i found and i got $f(5,5)=50$.

How can I know if this is the minimum or the maximum.

Answer 1

Using directly the constraint $g(x,y)=0$, we have that

$$h(x)=f(x,10-x)= x^2 + (10-x)^2=2x^2-20x+100$$

with

• $h'(x)=4x-20=0 \implies x=5$

and

• $h''(x)=4>0$

which means we are dealing with a minimum.

Answer 2

There are many ways to do this one. Inequality is fastest ! here is one: $x^2+y^2 \ge \dfrac{(x+y)^2}{2}= \dfrac{10^2}{2} = 50$ and equality occurs when $x = y = 5$. Hope this helps...