Using Lagrange Multipliers, find the max and min of $$x_1 y_1 + ...... + x_n y_n$$ subject to constraints $$x_1^p + ...... + x_n^p = X^p$$ $$y_1^q + ...... + y_n^q = Y^q$$ with $\frac1p + \frac1q = 1$, (assuming $x_i \geq 0$ and $y_i > 0$ for all i).
I know the objective function $f(x) = x_1 y_1 + ...... + x_n y_n$, but I have no clue on how to transform $\frac1p + \frac1q = 1$ to constraints with x and y to apply Lagrange. (All I can think of is $x_1^{\frac1p}x_1^{\frac1q} y_1^{\frac1p}y_1^{\frac1q} + ......+ x_n^{\frac1p}x_n^{\frac1q}y_n^{\frac1p}y_n^{\frac1q} = x_1 y_1 + ...... + x_n y_n$, but I don't think it's relevant). I'm new to optimization and lagrangian so I really don't know how to proceed. Can I have some hints? Thank you so much.