Step 1
What rule should we use?
This function is written as one expression divided by another.
Step 2
Identify \(u\) and \(v\)
For the quotient rule, write the function as \(\frac{u}{v}\).
Step 3
Differentiate \(u\) and \(v\)
If \(u=x^2+1\) and \(v=x\), what are \(u'\) and \(v'\)?
Step 4
Apply the quotient rule
Use \(\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{vu'-uv'}{v^2}\).
Step 5
Find where the derivative is zero
After simplifying, we get
\[
f'(x)=\frac{x^2-1}{x^2}
\]
For a stationary point, which part must equal zero?
Step 6
Solve for the stationary points
Set the numerator equal to zero and solve.
\[
x^2-1=0
\]
\[
(x-1)(x+1)=0
\]
Step 7
Final answer and graph
So, the stationary points happen at the x-values \(\,x=-1\,\) and \(\,x=1\).
\[
\text{Stationary points occur when } x=-1 \text{ and } x=1.
\]