If a student has a utility function given by $$U(x_1, x_2) = −x_1 + > 10x_ 2^2 − 2x_1x_2$$ where $X_1 = 5$ and $X_2 = 20$. If this student was to eat $5$ less hot meals per month, estimate the increase in the number of pints of beer the student will need to consume if they are to maintain their current utility.
I said that $$\frac{\partial U}{\partial X_1} = -1-2x_2 = -1 -2(20) =-41$$ $$\frac{\partial U}{\partial X_2} = 20x_2 -2x_1 = 20(20) - 2(5) = 390$$
When calculating that change in current utility, I start to run into trouble (I think).
$$ Δ = \frac{\partial U}{\partial X_1} Δx_1 + \frac{\partial U}{\partial X_2} Δx_2 $$ $$ 0 = (-41)(Δx_1) + (390)(5) $$ $$ \frac{1950}{-41} = Δx_1 $$ $$ Δx_1 = -47.56$$
So does that mean that the student needs to drink $47.56$ less beers to maintain the current utility? And is this possible when the student already only consumes $5$ beers to begin with?
Is $X_1$=amount of pints of beer ?
Then the equation is $0=-41\cdot \Delta x+390\cdot (\color{blue}{-5})$. The change is negative.
The result for $\Delta x$ is smaller than -5 as well. You are right. It is not possible to compensate the 5 hot meals with beer.