Explanation
To solve the question, we will make use of the graph of the function
[tex]f(x)=9x+2x^{-1}[/tex]
The graph of the function is shown below
From the graph
Looking at a graph, the local maxima and minima are the points where the graph flattens out and changes from increasing to decreasing or vice versa. When the graph is flat, that means the slope is zero.
[tex]\mathrm{Extreme\:Points\:of}\:9x+2\cdot \frac{1}{x}:\quad \mathrm{Maximum}\left(-\frac{\sqrt{2}}{3},\:-6\sqrt{2}\right),\:\mathrm{Minimum}\left(\frac{\sqrt{2}}{3},\:6\sqrt{2}\right)[/tex]
Therefore, the function has a local maximum at
[tex]x=\frac{-\sqrt{2}}{3}[/tex]
with a value of
[tex]-6\sqrt{2}[/tex]
For the local minimum
The function has a local minimum at
[tex]x=\frac{\sqrt{2}}{3}[/tex]
with a value of
[tex]6\sqrt{2}[/tex]
