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(x) = x/lnx

NB: f(x) is defined for x in RR >0, x!=1, so all values of x will be positive. Also, f(x) -> -oo as x-> 1 from below and f(x)-> +oo as x-> 1 from above.

To find a turning point set f'(x) =0

f'(x) = (lnx * 1 - x*1/x)/[lnx]^2 = 0

lnx-1=0

lnx=1

x=e

Hence f(x) has a turning point at x=e

Now observe the graph of f(x) below:

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

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

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