How to verify this optimization solution is correct?

Question

We're wanting to find the maximum of the production function

$f(x,y) = x^{0.7}y^{0.3}$ subject to budget constraint $2x + 4y = 3.$

Using Lagrange multiplier, I found $x = 1.05, \ y = 0.2247.$

Is this correct? How to verify?

Answer 1

Hint: since this is a budget constraint, you are implicitly given the constraints $x \geq 0$ and $y \geq 0$. Once you check the value of $f(x,y)$ on $x=0$ and $y=0$ and see that it is no bigger than $f(1.05, 0.2247)$, then you have proved that it is indeed the maximum.