Function Expression :
\[f(x)=x(\sqrt{x}-2 )^{25} \]Domain
\[\left[0, \infty\right[ \]Limits
\[\lim_{x \overset{>}{\rightarrow0} }f(x) = 0 \]\[\lim_{x \rightarrow+\infty}f(x) = +\infty \]
\[ \]
Derivate
\[f^{\,\prime}(x)=\frac{25 \sqrt{x} \left(\sqrt{x} - 2\right)^{24}}{2} + \left(\sqrt{x} - 2\right)^{25} \]\[f^{\,\prime}(x)=\frac{\left(\sqrt{x} - 2\right)^{24} \cdot \left(27 \sqrt{x} - 4\right)}{2} \]
\[ \]
Integral
\[F(x) = \frac{2 x^{\frac{29}{2}}}{29} + \frac{800 x^{\frac{27}{2}}}{9} + 16192 x^{\frac{25}{2}} + 985600 x^{\frac{23}{2}} + \frac{184588800 x^{\frac{21}{2}}}{7} + 352337920 x^{\frac{19}{2}} + 2505932800 x^{\frac{17}{2}} + 9737338880 x^{\frac{15}{2}} + \frac{267776819200 x^{\frac{13}{2}}}{13} + 22911385600 x^{\frac{11}{2}} + \frac{37140561920 x^{\frac{9}{2}}}{3} + \frac{19293798400 x^{\frac{7}{2}}}{7} + 167772160 x^{\frac{5}{2}} - \frac{25 x^{14}}{7} - \frac{18400 x^{13}}{13} - 141680 x^{12} - 5593600 x^{11} - 104600320 x^{10} - \frac{3042918400 x^{9}}{3} - 5325107200 x^{8} - \frac{107110727680 x^{7}}{7} - 23627366400 x^{6} - 18570280960 x^{5} - 6632243200 x^{4} - 838860800 x^{3} - 16777216 x^{2} \]Sign Table
Variation Table
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