Answer:
Analyzed and Sketched.
Step-by-step explanation:
We are given [tex]y=\frac{\ln\left(5x)}{x^2}[/tex]
To sketch the graph we need to find 2 components.
1) First derivative of y with respect to x to determine the interval where function increases and decreases.
2) Second derivative of y with respect to x to determine the interval where function is concave up and concave down.
[tex]y'=\frac{1-2\ln\left(5x)}{x^3}=0[/tex]
[tex]x = \sqrt e/5[/tex] is absolute maximum
[tex]y''=\frac{6\ln\left(5x)-5}{x^4}=0[/tex]
[tex]x=e^{5/6}/5[/tex] is the point concavity changes from down to up.
Here, x = 0 is vertical asymptote and y = 0 is horizontal asymptote.
The graph is given in the attachment.
