extreme f(x)=-5x^{5/3}+5\sqrt[3]{x}

Lösung

extrempunkte

f (x) = - 5 x^{\frac{5}{3}} + 5 3 x

Minimum (- \frac{1}{5 ^{\frac{3}{4}}}, - 5 (- \frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{1}{3}}), Maximum (\frac{1}{5 ^{\frac{3}{4}}}, - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} + 5 3 \frac{1}{5 ^{\frac{3}{4}}})

Schritte zur Lösung

f^{'} (x) = - \frac{25 x ^{\frac{2}{3}}}{3} + \frac{5}{3 x ^{\frac{2}{3}}}

Intervalle finden:

Absteigend

: - \infty < x < - \frac{1}{5 ^{\frac{3}{4}}},

Ansteigend

: - \frac{1}{5 ^{\frac{3}{4}}} < x < 0,

Ansteigend

: 0 < x < \frac{1}{5 ^{\frac{3}{4}}},

Absteigend

: \frac{1}{5 ^{\frac{3}{4}}} < x < \infty

Stecke

x = - \frac{1}{5 ^{\frac{3}{4}}}

- 5 x^{\frac{5}{3}} + 5 3 x : - 5 (- \frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{1}{3}}

Minimum (- \frac{1}{5 ^{\frac{3}{4}}}, - 5 (- \frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{1}{3}})

Stecke

x = \frac{1}{5 ^{\frac{3}{4}}}

- 5 x^{\frac{5}{3}} + 5 3 x : - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} + 5 3 \frac{1}{5 ^{\frac{3}{4}}}

Maximum (\frac{1}{5 ^{\frac{3}{4}}}, - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} + 5 3 \frac{1}{5 ^{\frac{3}{4}}})

Minimum (- \frac{1}{5 ^{\frac{3}{4}}}, - 5 (- \frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{1}{3}}), Maximum (\frac{1}{5 ^{\frac{3}{4}}}, - 5 (\frac{1}{5 ^{\frac{3}{4}}})^{\frac{5}{3}} + 5 3 \frac{1}{5 ^{\frac{3}{4}}})

e x t re m e - x^{2} + 100 x + 8

e x t re m e y = x 16 - x^{2}

e x t re m e f (x, y) = 7 e^{y} - 6 y e^{x}

e x t re m e f (x) = \frac{5}{x - 2}

e x t re m e y = x^{2} - 15 x + 54