Let $f(x)=\begin{cases} \frac{a\log x}{5(x-1)}&;x>1\\ \\ -\frac{bx^2}5+x&;x\leq 1. \end{cases}$
I would like to find $a$ and $b$, so that the function will be continuous and have a derivative at 1.
Lets say that $f_{R}(x) = \frac{a\log(x)}{5(x-1)}$ and $f_{L}(x) = -\frac{bx^2}5+x$.
The first thing I did was to find the limit of $f_{R} = \frac{a}{5}$ and $f_{L}=\frac{5-b}{5}$. The function will be continuous, if these two limits will be equal.
How would I proceed and how would I find a and b so that it would fit the criteria at the beginning of this post?