I am new to Lagrangians, and I am not sure if what I am doing is correct. The original problem was to find $\min\limits_{\theta}-log(\theta(1-\theta)^2), .5 \leq \theta \leq 1$, write the Lagrangian, and to derive the dual problem. My solution is $\theta = 0.5$.
My Lagrangian is $\mathcal{L}(\theta, \lambda) = -log(\theta(1-\theta)^2) - \lambda(\theta-0.5)$, $\lambda \geq 0$.
I constructed the dual function $g(\lambda) = \min\limits_{\theta} \mathcal{L}(\theta, \lambda) = \min\limits_{\theta} -log(\theta(1-\theta)^2) - \lambda(\theta-0.5), \lambda \geq 0$. My solution is $\theta = 0.5, \lambda = 2$.
I am not completely sure if my work is correct, and I am not sure how to construct the dual problem. My best attempt is $d^* = \max\limits_{\lambda \geq 0} g(\lambda) $
Is my work correct? How do you express the dual problem?