Question

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

Derivatives
ANSWERED
asked 2021-06-08
Use derivatives to find the critical points and inflection points. \(\displaystyle{f{{\left({x}\right)}}}={5}{x}−{3}{\ln{{x}}}\)

Answers (1)

2021-06-09
\(\displaystyle{f{{\left({x}\right)}}}={5}{x}-{3}{\ln{{x}}}\)
\(\displaystyle{f}'{\left({x}\right)}={5}-{\frac{{{3}}}{{{x}}}}={\frac{{{5}{x}-{3}}}{{{x}}}}\)
\(\displaystyle{f}{''}{\left({x}\right)}={\frac{{{3}}}{{{x}^{{{2}}}}}}\)
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.
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