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

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

Or as an inequality:

Step-by-step explanation:

We are given the equation:

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:

Differentiate:

y is decreasing whenever y' is negative. Thus:

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

Factor:

eˣ is always positive. So:

Adding one to both sides produces:

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

In interval notation:

Or as an inequality:

step-by-step explanation:

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