Answer:
The interval for which <em>y</em> is a decreasing function of <em>x</em> is:
Or as an inequality:
Step-by-step explanation:
We are given the equation:
And we want to find the range of values <em>x</em> for which <em>y</em> is a decreasing function of <em>x</em>.
<em>y</em> is decreasing whenever <em>y'</em> is negative. Find <em>y'</em> using the Quotient Rule:
Differentiate:
<em>y</em> is decreasing whenever <em>y'</em> is negative. Thus:
Multiply both sides by <em>x²</em>. This is always positive so we do not need to change the sign:
Factor:
e<em>ˣ</em> is always positive. So:
Adding one to both sides produces:
Therefore, <em>y</em> is a decreasing function of <em>x</em> when <em>x</em> is less than one (and greater than 0).
In interval notation:
Or as an inequality: