<s>[INST] If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ Can some one help me out on where to go?
If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ ?
 [/INST] Your problem is as follows:
\begin{align*}
&\max \,2a+b\\
\\
&\text{s.t. } a^2 + b^2=1
\end{align*}
Using substitution, let $f(b) = 2a + b = 2\sqrt{1 - b^2} +b$. Now, we wish to find the maximum of this function with respect to $b$.
Using simple derivatives, we get
$$
\frac{-2b}{\sqrt{1-b^2}} + 1 = 0\implies 2b=\sqrt{1-b^2} \implies 5b^2=1\implies b=\sqrt{1/5}\implies a = \sqrt{4/5}.
$$
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