Question

Given that y = (e^x)/x, x > 0, find the range of values of x where y is a decreasing

Answer 1

Answer:

The interval for which y is a decreasing function of x is:

$(0, 1)$

Or as an inequality:

$0$

Step-by-step explanation:

We are given the equation:

$\displaystyle y=\frac{e^x}{x}\, ,x>0$

And we want to find the range of values x for which y is a decreasing function of x.

y is decreasing whenever y' is negative. Find y' using the Quotient Rule:

$\displaystyle y'=\frac{(e^x)'(x)-e^x(x)'}{(x)^2}$

Differentiate:

$\displaystyle y'=\frac{xe^x-e^x}{x^2}$

y is decreasing whenever y' is negative. Thus:

$\displaystyle 0>\frac{xe^x-e^x}{x^2}$

Multiply both sides by x². This is always positive so we do not need to change the sign:

$xe^x-e^x$

Factor:

$e^x(x-1)$

eˣ is always positive. So:

$x-1$

Adding one to both sides produces:

$x$

Therefore, y is a decreasing function of x when x is less than one (and greater than 0).

In interval notation:

$(0, 1)$

Or as an inequality:

$0$