Maximum value of function

Question

I was exercising calculus when I stumbled upon this question:

Let $f$ be the function defined by $$f(x)=\frac {\ln x}{x}$$

What is the absolute maximum value of $f$?

1. $1$

2. $\cfrac1e$

3. $0$

4. $-e$

5. $f$ does not have an absolute maximum value

In this section I was not supposed to use a calculator so I couldn't have solved for critical points as

$$f'(x) = \frac{1-\ln x}{x^2}$$

So if you could help me out, I would greatly appreciate it. Thank you.

Answer 1

You correctly found the derivative to be: $$f'(x)=\frac{1-\ln(x)}{x^2}$$ To find local extrema, let $f'(x)=0$. $$\frac{1-\ln(x)}{x^2}=0$$ If we multiply both sides by $x^2$, we reduce the problem to: $$1-\ln(x)=0$$ Can you continue?

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You can verify that your point is a maximum at a point $(c,f(c))$ by checking if $f''(c)<0$. You can also verify that this point is not only a local maximum, but also an absolute maximum.