I encounter this problem when reading the proof of Bernstein inequality for bounded random variables.
Consider the function $h(\lambda) = \frac{\lambda^2\cdot v^2/2}{1-c\lambda}-\lambda t$ for some fixed $v^2,c,t>0$ and $v^2\geq c^2$. If we want to minimise $h(\lambda)$ with the constraint $\lambda c<1$. Then if we consider the optimal $\lambda$, it should be $$\lambda = \frac{1}{c}(1-\frac{v}{\sqrt{v^2+2ct}}),$$ with the optimal value $\frac{v^2}{c^2}(\sqrt{1+\frac{2t}{c}}-1)-\frac{t}{c}$. However, in the proof, the authors just take $\lambda = \frac{t}{v^2+ct}$ and the corresponding value is $-\frac{t^2/2}{v^2+ct}$.
My question is what is the magic of choosing $\lambda$ here so that the result is much more neat than the actual minimum. Thanks in advance.