extreme y=x^{2/7}(x^2-5)

Lösung

extrempunkte

y = x^{\frac{2}{7}} (x^{2} - 5)

Minimum - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 ( - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} ) ^{\frac{2}{7}}}{8}, Maximum (0, 0), Minimum (\frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 \cdot 5 ^{\frac{1}{7}}}{8 \cdot 2 ^{\frac{3}{7}}})

Schritte zur Lösung

f^{'} (x) = \frac{2 ( 2 ^{1 + \frac{1}{2}} x + 5 ^{\frac{1}{2}} ) ( 2 ^{1 + \frac{1}{2}} x - 5 ^{\frac{1}{2}} )}{7 x ^{\frac{5}{7}}}

Intervalle finden:

Absteigend

: - \infty < x < - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}},

Ansteigend

: - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} < x < 0,

Absteigend

: 0 < x < \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}},

Ansteigend

: \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} < x < \infty

Stecke

x = - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}

x^{\frac{2}{7}} (x^{2} - 5) : - \frac{35 ( - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} ) ^{\frac{2}{7}}}{8}

Minimum - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 ( - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} ) ^{\frac{2}{7}}}{8}

Stecke

x = 0

x^{\frac{2}{7}} (x^{2} - 5) : 0

Maximum (0, 0)

Stecke

x = \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}

x^{\frac{2}{7}} (x^{2} - 5) : - \frac{35 \cdot 5 ^{\frac{1}{7}}}{8 \cdot 2 ^{\frac{3}{7}}}

Minimum (\frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 \cdot 5 ^{\frac{1}{7}}}{8 \cdot 2 ^{\frac{3}{7}}})

Minimum - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 ( - \frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}} ) ^{\frac{2}{7}}}{8}, Maximum (0, 0), Minimum (\frac{5 ^{\frac{1}{2}}}{2 \cdot 2 ^{\frac{1}{2}}}, - \frac{35 \cdot 5 ^{\frac{1}{7}}}{8 \cdot 2 ^{\frac{3}{7}}})

e x t re m e y = - t u (t - 1) + t u (t - 2)

e x t re m e f (x) = 2 x^{2} + 2 y^{2} - 8 x + 16 y

e x t re m e f (x) = x^{2} + y^{2} - 8 y + 16

e x t re m e x^{3} (x - 1)^{2} (7 x - 4)

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