1. ## Maximum Value

Find the maximum value of:

$\displaystyle f(x) = x^2 ln(1/x)$

2. $\displaystyle \frac{d}{dx} x^{2}\ln \Big(\frac{1}{x}\Big) = 2x \ln \Big(\frac{1}{x}\Big) +x^{2}*\frac{1}{1/x}\Big(\frac{-1}{x^{2}}\Big) = 2x \ln \Big(\frac{1}{x} \Big) -x$

$\displaystyle 2x \ln \Big(\frac{1}{x} \Big) -x = 0$

$\displaystyle \ln \Big(\frac{1}{x} \Big) = \frac{1}{2}$

$\displaystyle x = \frac{1}{e^{1/2}}$

3. Originally Posted by penguinpwn
Find the maximum value of:

$\displaystyle f(x) = x^2 ln(1/x)$
Make life a bit easier by noting that $\displaystyle f(x) = - x^2 \ln (x)$. Now apply the product rule.

4. Originally Posted by mr fantastic
Make life a bit easier by noting that $\displaystyle f(x) = - x^2 \ln (x)$. Now apply the product rule.
Does that property of ln work with all inverses?

for example, would $\displaystyle f(x) = ln (cscx)$be equal to $\displaystyle f(x) = ln (sinx)$?

5. Originally Posted by penguinpwn
Does that property of ln work with all inverses?

for example, would $\displaystyle f(x) = ln (cscx)$be equal to $\displaystyle f(x) = ln (sinx)$?
You should have learned in pre-calculus that $\displaystyle \log \frac{1}{A} = - \log A$.