Function Expression :
\[f(x)=\frac{-1}{2}x+3+\frac{ln x}{2x} \]Domain
\[\left]0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = -\infty \]\[\lim_{x \rightarrow+\infty}f(x) = -\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{\frac{1}{2} \cdot \frac{1}{x}}{x} - \frac{1}{2} - \frac{\log{\left(x \right)}}{2 x^{2}} \]\[f^{\,\prime}(x)=\frac{- x^{2} - \log{\left(x \right)} + 1}{2 x^{2}} \]
\[ \]
Integral
\[F(x) = - \frac{x^{2}}{4} + 3 x + \begin{cases} 0 & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \wedge \left|{x}\right| < 1 \\\frac{\log{\left(x \right)}^{2}}{4} & \text{for}\: \left|{x}\right| < 1 \\\frac{\log{\left(\frac{1}{x} \right)}^{2}}{4} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\\frac{{G_{3, 3}^{3, 0}\left(\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle| {x} \right)}}{2} + \frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle| {x} \right)}}{2} & \text{otherwise} \end{cases} \]Sign Table
Variation Table
Plot
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