Function Expression :
\[f(x)=-\frac{2ln x}{3x}-\frac{1}{x} \]Domain
\[\left]0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = +\infty \]\[\lim_{x \rightarrow+\infty}f(x) = 0 \]
\[ \]
Derivate
\[f^{\,\prime}(x)=- \frac{2 \cdot \frac{1}{3 x}}{x} + \frac{2 \log{\left(x \right)}}{3 x^{2}} + \frac{1}{x^{2}} \]\[f^{\,\prime}(x)=\frac{2 \log{\left(x \right)} + 1}{3 x^{2}} \]
\[ \]
Integral
\[F(x) = - \begin{cases} 0 & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \wedge \left|{x}\right| < 1 \\\frac{\log{\left(x \right)}^{2}}{3} & \text{for}\: \left|{x}\right| < 1 \\\frac{\log{\left(\frac{1}{x} \right)}^{2}}{3} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\\frac{2 {G_{3, 3}^{3, 0}\left(\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle| {x} \right)}}{3} + \frac{2 {G_{3, 3}^{0, 3}\left(\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle| {x} \right)}}{3} & \text{otherwise} \end{cases} - \log{\left(x \right)} \]Sign Table
Variation Table
Plot
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