Let $a_j, c_j , j=1,...n$ and b be positive constants. $$-$$ Minimize : $\sum_{j=1}^n \frac{c_j}{x_j}$ $$-$$ subject to:$\sum_{j=1}^n a_j x_j =b$
$$-$$
Write down the kkt conditions and solve the resulting equations\begin{align} \\ & f(x) = \sum_{j=1}^n \frac{c_j}{x_j} \\ & h(x)= \sum_{j=1}^n a_j x_j -b \\ & \nabla f(x) = -\sum_{j=1}^n \frac{c_j}{x_j^2} \\ & \nabla h(x) = \sum_{j=1}^n a_j \end{align} $\nabla L(x,v) = -\sum_{j=1}^n \frac{c_j}{x_j^2} + v*(\sum_{j=1}^n a_j ) = 0$
And we look for the cases when v = 0 and v is free? v could not be zero due to the fact that our x goes to infinity so x has to be < or > 0. My last step seems to be wrong, is the derivatives wrong? Thankful for any kind of hint or help.