utility function question from my textbook

Question

Suppose there are two goods with prices $ p₁ = 2, p₂ = 5, $ the income is $ M = 40 $ and the utility function is $ U (x₁, x₂) = (x₁)^⅓ . (x₂)^ ½, $ Find the optimum consumption plan.

Attempt:

I do not understand how to do this. The examples in my textbook are not explanatory enough. They are kinda complex. Can someone please help out with this question. The textbook says the final answer is $ MRS (x₁, x₂)= 2y/3x $

Its not an assignment question by the way. Just solving random questions.

Answer 1

The consumer is solving $$\max x_1^{1/3}x_2^{1/2}$$ subject to $$2 x_1 + 5 x_2 \leq 40.$$

It should be clear that the constraint must bind ($2x_1 +5x_2=40$), since the objective is increasing in both $x_1$ and $x_2$. Using the method of lagrange multipliers, any optimal consumption plan must satisfy the following FOCs: $$1/3 x_1^{-2/3} x_2^{1/2}=2 \lambda,$$ $$1/2 x_1^{1/3} x_2^{-1/2}=5\lambda.$$ Rearranging terms gives us $$x_1=\frac{5}{3} x_2.$$ Finally, plugging this back into the constraint, we get $$2x_1+3 x_1=40$$ So $x_1=8$, $x_2=24/5$.

Answer 2

We are asked to maximize $U(x_1,x_2)$ subject to $2x_1+5x_2=40$. We can rearrange the constraint to get $x_1=\frac 12(40-5x_2)$. We might as well maximize $U^3(x_1,x_2)=x_1x_2=\frac 12(40-5x_2)x_2$. Now take the derivative, set to zero, and you will have the value for $x_2$.