Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ It is given that $4 x^4 + 9 y^4 = 64$.
Then what will be the maximum value of $x^2 + y^2$?
I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .

Since you tagged Lagrange Multipliers, I will use that method.
Allow that $F(x,y) = 4x^4+9y^4-64$ and $G(x,y)=x^2+y^2-C$, where $C$ is treated as a constant and is the maximum you wish to find.
We allow that $\nabla F(x,y) = \lambda \nabla G(x,y)$ and find two new equations:
$16x^3 = \lambda 2x$ and
$36y^3 = \lambda 2y$
$\implies x^4 = \frac{\lambda ^2}{64}$ and $y^4 = \frac{\lambda ^2}{324}$.
Plugging those into $F(x,y)$ we find that:
$$\lambda = 26.6256...$$
We now go back to our equations after applying the gradient and solve for $x_{max}$ and $y_{max}$. We get:
$x_{max}=1.8245...$ and $y_{max}=1.21622...$
Plugging this into $x^2+y^2$ we obtain the maximum, $4.80727$.
