Let f(x)=ln(x)/sqrtx. Find the critical values of f(x).

Let $f\left(x\right)=\frac{\mathrm{ln}\left(x\right)}{\sqrt{x}}$. Find the critical values of f(x)
How would you find the critical value of this equation? So far... I have gotten that you find the derivative of f(x) which is ${f}^{\prime }\left(x\right)=\frac{1}{{x}^{3/2}}-\frac{\mathrm{ln}\left(x\right)}{2{x}^{3/2}}.$.
To find the critical values set ${f}^{\prime }\left(x\right)=0$? But how would you factor out the derivative to find the critical values?
Also, after that how would you find the intervals of increase and decrease and relative extreme values of f(x).
And lastly the intervals of concavity and inflection points.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

wayiswa8i
Explanation:
Since ${f}^{\prime }\left(x\right)=\frac{2-\mathrm{ln}x}{{x}^{3/2}}$, ${f}^{\prime }\left(x\right)=0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}x={e}^{2}$. Furthermore, ${f}^{\prime }\left(x\right)<0$ when $x>{e}^{2}$ and ${f}^{\prime }\left(x\right)>0$ when $0
Did you like this example?
Julia Chang
Explanation:
Rewrite f′ as ${f}^{\prime }\left(x\right)=\frac{1-1/2\mathrm{ln}x}{{x}^{3/2}}$ and to find ${f}^{\prime }=0$ you need to find when $1-1/2\mathrm{ln}x=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{ln}x=2$.
Did you like this example?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee