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 $2a + b$ ?

We are given that $ a^2 + b^2 = 1$, and $ a, b \geq 0$.
Applying AM-GM or Cauchy-Schwarz, we get that
$$ 20 = 5(a^2 + a^2 + a^2 + a^2 + 4b^2) \geq (a+a+a+a+2b)^2 = (4a+2b)^2$$
Hence $ 2a + b \leq \frac{20}{16} = \frac{5}{4} $, with equality when $ a = 2b, b = \sqrt{\frac{1}{5} }$.
