extreme x^{1/3}(x^2-9)

Lösung

extrempunkte

x^{\frac{1}{3}} (x^{2} - 9)

Maximum (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Minimum (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

Schritte zur Lösung

f^{'} (x) = \frac{( 7 ^{\frac{1}{2}} x + 3 ) ( 7 ^{\frac{1}{2}} x - 3 )}{3 x ^{\frac{2}{3}}}

Intervalle finden:

Ansteigend

: - \infty < x < - \frac{3}{7 ^{\frac{1}{2}}},

Absteigend

: - \frac{3}{7 ^{\frac{1}{2}}} < x < 0,

Absteigend

: 0 < x < \frac{3}{7 ^{\frac{1}{2}}},

Ansteigend

: \frac{3}{7 ^{\frac{1}{2}}} < x < \infty

Stecke

x = - \frac{3}{7 ^{\frac{1}{2}}}

x^{\frac{1}{3}} (x^{2} - 9) : \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}

Maximum (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

Stecke

x = \frac{3}{7 ^{\frac{1}{2}}}

x^{\frac{1}{3}} (x^{2} - 9) : - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}

Minimum (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

Maximum (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Minimum (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

e x t re m e f (x) = x \cdot e^{x}

e x t re m e f (x) = x^{3} + 2 x^{2} - 20 x, 0 \leq x \leq 3

e x t re m e f (x) = - x^{3} + 3 x - 1

e x t re m e x^{3} - 9 x + 2

e x t re m e f (x) = 2 x^{3} - 9 x^{2} - 60 x + 8