General technique for solving these optimization problems

There's three particular optimization problems I see quite often(in various applications). They seem to me to be specific cases of a more general problem, for which there perhaps exists a technique with which they can be solved.

1) Suppose $x_{1}...x_{n}$ are positive reals such that $x_{1}+...+x_{n} = K$. Then maximize the product $x_{1}*x_{2}*...*x_{n}$.

2) Suppose $x_{1}...x_{n}$ are positive reals such that $x_{1}*...*x_{n} = K$. Then minimize the sum $x_{1}+x_{2}+...+x_{n}$.

3) Suppose $p_{1}...p_{n}$ are positive reals, each less than or equal to 1, i.e. $0 < p_{i} \leq 1 \$. Then minimize $p_{1}*log(p_{1})+...+p_{n}*log(p_{n})$ (this number is going to be negative, so make it as negative as possible.)

How could I solve these three problems? I'm versed in calculus, and also analysis, but can't seem to think of an appropriate technique.

I was reading about convex functions, what bearing do they have on the problem? What about techniques from elementary calculus?

This type of optimisation problems can be solved using the method of Lagrange multipliers , a method for finding the local maxima and minima of a function subject to equality constraints. Given a function to maximise (minimise) $f(x,y)$ and a constraint $h(x,y)=0$ you write down an auxiliary function $\mathcal{L}(x,y,\lambda_1)=f(x,y)-h(x,y)\lambda_1$. Then, derive $\mathcal{L}(x,y,\lambda_1)$ wrt to $x,y,\lambda_1$, set the derivatives equal zero and solve the system in $x,y,\lambda_1$ to find the optimal values $(x^*,y^*)$ satisfying the constraint and the minimisation/maximization problem. You can generalise it to the case of more variables and/or constraints.