Question

# Use derivatives to find the critical points and inflection points. f(x)=5x−3\ln x

Derivatives
Use derivatives to find the critical points and inflection points. $$\displaystyle{f{{\left({x}\right)}}}={5}{x}−{3}{\ln{{x}}}$$

$$\displaystyle{f{{\left({x}\right)}}}={5}{x}-{3}{\ln{{x}}}$$
$$\displaystyle{f}'{\left({x}\right)}={5}-{\frac{{{3}}}{{{x}}}}={\frac{{{5}{x}-{3}}}{{{x}}}}$$
$$\displaystyle{f}{''}{\left({x}\right)}={\frac{{{3}}}{{{x}^{{{2}}}}}}$$
For critical points, we set f'(x) = 0 and we get x = $$\displaystyle{\frac{{{3}}}{{{5}}}}$$ For inflection points, since f''(x) is not zero anywhere and undefined at point x=0, it is candidate for point of inflection. Since the function f(x) itself is undefined at this point ($$\displaystyle{\ln{{0}}}$$ = undefined), it is not an inflection point. There are no inflection points here.