<s>[INST] The maximum and minimum of a function How can I find the maximum and minimum of $$L=\sqrt{1-x^2}+\sqrt{1-y^2}+\sqrt{1-z^2},$$
given that $x^2+y^2+z^2=1$?
I worked on some basic algebra, but it doesn't get me anywhere, so I'm stuck on it.
 [/INST] By Cauchy-Schwarz inequality
$$\sum_{cyc}\sqrt{1-x^2}\le\sqrt{3\cdot(1-x^2+1-y^2-z^2)}=\sqrt{6}$$
By the other hand,we have
\begin{align*}
&\sqrt{1-x^2}+\sqrt{1-y^2}+\sqrt{1-z^2}=\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{x^2+z^2}\\
&\ge\sqrt{0+y^2}+\sqrt{0+z^2}+\sqrt{y^2+z^2}\\
&=|y|+|z|+\sqrt{y^2+z^2}
\end{align*}
this problem since $y^2+z^2=1$,find the $|y|+|z|+1$ minum
let
$$|y|=a\ge 0,|z|=b\ge0, a^2+b^2=1$$
then
$$|y|+|z|+1=a+b+1=\cos{t}+\sin{t}+1=\sqrt{2}\sin{(t+\dfrac{\pi}{4})}+1\ge 2$$
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