Find the values of 'a' for which $\displaystyle f(x)=x^{2}-2 a x+b$ is an increasing function for $\displaystyle x>0$.

\[\begin{aligned} & f^{\prime}(x)=2 x-2 a \\ & 0<x<\infty \Rightarrow-2 a<2 x-2 a<\infty \Rightarrow-2 a<f^{\prime}(x)<\infty \\ & f(x) \text { is increasing iff } f^{\prime}(x) \geq 0 \\ & \Rightarrow-2 a \in[0, \infty) \Rightarrow a \in(-\infty, 0] \text { or }(-\infty, 0) \end{aligned} \]