Nonlinear optimization using Lagrange Duality

Question

I've been trying to solve the following exercise: I need to maximize the function $$ f(x_1, x_2, x_3) = x_1^2 + 6x_2 + 4x_3^2 $$ with constraints $$ 2x_1 + x_2 + 4x_3 \leq 10\\ x_1, x_3 \geq 0\\ x_2 \in [0, 1] $$ using the Lagrangian Dual method. I used the following example https://i.stack.imgur.com/uB9Jy.jpg and tried to replicate it on my problem. Firstly, I changed the maximization into minimization by considering $-f(x_1, x_2, x_3)$. Then formed the Lagrangian: $$ L(x_1, x_2, x_3) = -x_1^2 - 6x_2 - 4x_3^2 + \lambda (2x_1 +x_2 + 4x_3 - 10). $$ Now the Lagrangian dual function is, after grouping same variables: \begin{align} \theta(\lambda) &= \min_{x_1 \geq 0} \{-x_1^2 + 2\lambda x_1\}\\ &+\min_{x_2\in [0, 1]} \{-6x_2 + \lambda x_2\} \\ &+ \min_{x_3 \geq 0} \{-4x_3^2 + 4\lambda x_3\} - 10\lambda. \end{align} For the next step I need to resolve these minima in the function but they have negative coefficients at the quadratic member and I cannot resolve them.