A company produces one good using two factors of production factors. If $x$ and $y$ denotes the units of the factors used by the company, the technology function is given by $F(x; y) = xy^2$. In the market of the production factors the price of the factor $x$ is 5 but the price of $y$ is unknown. However we can observe that when the total production of the company is $Q = 1000$ units the company uses 10 units of $x$ and 10 units of $y$.
(a) If the company acts rationally (it wants to minimize the cost of the production) which is the price of the factor $y$? (b) Which is the marginal increment of the cost of production if the company decides to (marginally) increase the production?
First you should convince yourself that the inputs that firm chooses satisfy $xy^2=1000$. If not, i.e. if $xy^2>1000$ then the firm could reduce cost by choosing an $x$ smaller than 10, therefore coming up with an alternative plan $(\tilde{x},\tilde{y})$ that has a smaller cost than $(10,10)$ but that it still guarantees an output at least 1000. Once we establish this we can replace $x=\frac{1000}{y^2}$. This means the firm chooses only $y$ (i.e. by choosing $y$ it implicitly chooses $x$).
Problem is $min \{p_x \frac{1000}{y^2}+p_y y\}$. Here $p_x=5$. First order conditions tell you that $p_y=\frac{10000}{y^3}$. Since $y$ is the optimized value, it is $10$, hence $p_y$is 10.
The answer to your last question is given by the Lagrange multiplier, which after you know prices, you find it as $\lambda=.05$