How to find the maximum value of the given set?

Question

Maximum $\left\{x+y \mid (x,y) \in \overline{B(0,1)}\right\}$ is equal to?

Answer 1

Note that your set consists of the couples $(x,y)$ with $x^2 + y^2 \leq 1$.

Clearly we'll want $x$ and $y$ to be positive, so we can rewrite the inequality as $$y \leq \sqrt{1-x^2}$$ Since we want to maximize the sum, it seems natural to then try $$y = \sqrt{1-x^2}$$ We then have to maximize the function $$f(x) = x + \sqrt{1-x^2}$$ The derivative is $$f'(x) = 1 - \frac{x}{\sqrt{1-x^2}}$$ which is zero when $x = \frac{1}{\sqrt{2}}$. Then $y$ is also $\frac{1}{\sqrt{2}}$.

Hence the maximum of the set is $$\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}$$

Answer 2

I draw the plot by wolframalpha :

Where $x+y=k$, you can see the $y$-axis which is the value $k$.