I'm trying to read a certain text, where they reach a computational complexity depending on scalars $a,b,c$ and a parameter $\lambda >0$ $$ O\left(\left\lceil\sqrt{\lambda a + \lambda^2 b^2} \right\rceil \max \left\{1, \frac{c}{\lambda}\right\}\right) \quad (1)$$ and try to choose the parameter $\lambda$ to minimize it.
Then they say it is straightforward to see that if $$\lambda_1 := \max\left\{\frac{a}{b^2}, \frac{1}{b}\right\} \leq c := \lambda_2$$ then any $\lambda \in [\lambda_1,\lambda_2]$ minimizes $(1)$ with $$O(1 + bc)$$ And otherwise, if $\lambda_2 < \lambda_1$ then $\lambda = \lambda_2$ minimizes $(1)$ with $$O(1 + \sqrt{ac}).$$ How is this straightforward? No matter what I try I'm not being able to get any of those two statements. How can I obtain those results?