Minimise the function $f(x) = \frac{x^2 - x +4}{x-1}$ using Calculus

Question

I had to minimise the function

$$f(x) = \frac{x^2 - x +4}{x-1}$$

I did the method where I found the range of this function and found the minimum value. However I know some basic calculus and was trying to find it using that but I am not able to. So, how do we find minima of this expression using calculus?

Answer 1

HINT

First of all, notice that

\begin{align*} f(x) = \frac{x^{2} - x + 4}{x - 1} = \frac{x^{2} - x}{x-1} + \frac{4}{x-1} = x + \frac{4}{x-1} \end{align*}

Then determine for which values of $x$ one has that $f'(x) = 0$. After so, verify whether $f''(x) > 0$.

Answer 2

If $$f(x) = \frac{x^2 - x + 4}{x-1} = x + \frac{4}{x-1}$$ then $$f'(x) = 1 - \frac{4}{(x-1)^2} \text{ and }f''(x) = \frac{8}{(x-1)^3}$$ Can you take it from here?