Determine whether $f(x)$ and $g(x)$ is decreasing or increasing without derivative

Question

Consider $f(x) = \frac{1}{\sqrt{x}} - x + 5$ and $g(x) = \frac{1}{\sqrt{x-1}} - \frac{1}{\sqrt{2-x}}$. We want to know that these functions are increasing or decreasing. I know that we can use determination sign of first derivative for checking it but I want the other ways to solve it. Also , one of the ways for doing it is that suppose $x_1 f(x_2)$ but I can't implement this method.If someone provide better ways or using this method is helpful.

Answer 1

Let us consider $0

Since the square root function is strictly growing, $\sqrt{x_1} < \sqrt{x_2}$ and since the inverse is strictly decreasing, $\frac{1}{\sqrt{x_1}} > \frac{1}{\sqrt{x_2}}$

Obviously, $-x_1 > -x_2$

By summing, we get $f(x_1)>f(x_2)$

I let you do the other one in a similar fashion (this time, take $1

Following comment, hint for $g$

$x \rightarrow \frac{1}{\sqrt{x-1}} $ is strictly decreasing on $]1,2[$

$x \rightarrow \frac{1}{\sqrt{2-x}} $ is strictly increasing on $]1,2[$

You can show these in a similar fashion