I have the following question about maximizing a function, and it makes complete intuitive sense in the way they went about solving the question.
But in part (c), when they say that $p_1=p_2=p_3=\dfrac 13$, it makes total sense. BUT if you plug in $1/3$ into any of the partial derivatives, it doesn't equal $0$? you get like $1.09-1$.
If instead you try to solve part (c) by solving for $p_1$,$p_2$, and $p_3$, individually, you get each one equals $1/e$. And when you do $p_1+p_2+p_3$, it equals $1.10$ (really similar to the other way). Why does this happen?
The partial derivatives would be equal to zero if you were maximizing over all $(p_1, p_2, p_3)$, but this is a constrained maximization problem, as we are only considering $(p_1, p_2, p_3)$ where $p_1 + p_2 + p_3 = 1$.
You can either use something like Lagrange multipliers, which is a general strategy, or, in this particular case, they were able to convert $H(p_1, p_2, p_3)$ to just be a function of $p_1$ and $p_2$.
I'll bet you were computing the partial derivative using the first expression, i.e.$$\frac{\partial}{\partial p_1} H(p_1, p_2, p_3)$$ when you were getting a non-zero value.
If you use the substituted expression (the one in part a), i.e. compute $$\frac{\partial}{\partial p_1} H(p_1, p_2)$$
We are maximizing it over all $(p_1, p_2)$ (well, all positive $(p_1, p_2)$) and I'll bet you get zero.