# The set of positive real values of x for which the function f(x) = x/ln x is a decreasing function is ?

## A x < e B x = 1 C x < e^2 D x > e E empty space

##### 1 Answer
Dec 3, 2016

A $x < e$

#### Explanation:

$f \left(x\right) = \frac{x}{\ln} x$

NB: $f \left(x\right)$ is defined for $x \in \mathbb{R} > 0 , x \ne 1$, so all values of x will be positive. Also, $f \left(x\right) \to - \infty$ as $x \to 1$ from below and $f \left(x\right) \to + \infty$ as $x \to 1$ from above.

To find a turning point set $f ' \left(x\right) = 0$

$f ' \left(x\right) = \frac{\ln x \cdot 1 - x \cdot \frac{1}{x}}{\ln x} ^ 2 = 0$

$\ln x - 1 = 0$

$\ln x = 1$

$x = e$

Hence $f \left(x\right)$ has a turning point at $x = e$

Now observe the graph of $f \left(x\right)$ below:

graph{x/lnx [-13.55, 17.64, -5.69, 9.9]}

It can be seen that $f \left(x\right)$ is decreasing for $x < e$ and increasing for $x > e$.

Hence the answer to this question is: A $x < e$