How would I go about solving this question:
Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. find:
a. The maximum amount of leisure the house hold can have and still pay taxes
b. find the optimal bundle of consumption and leisure.
For part (a), I took the derivative of $U$ with respect to $L$ and set it to 0, giving me $l^{0.5}=0$ and $l=0$. Is this part correct? Intuitively, this feels very wrong to me.
I'm also stuck on part (b). I took the derivative of the utility with respect to $C$. From my understanding, MRS is the derivative of $U$ with respect to $L$ over the derivative of $U$ with respect to $C$. I ended up getting $1/(L^{0.5})$.
I know the optimal bundle is when the budget line is tangent to the indifference curve and I also know that the slope of the budget line would be $-W$. However, when I solve for $1/(L^{0.5})=0.5$, I end up getting $L=4$ and $C=-1$. It doesn't make sense to me that a consumer will be able to have negative consumption.
Would greatly appreciate it if someone could point out where I went wrong!