extreme f(x)=x^5-x^3+1

Lösung

extrempunkte

f (x) = x^{5} - x^{3} + 1

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1), Sattel (0, 1), Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

Schritte zur Lösung

f^{'} (x) = 5 x^{4} - 3 x^{2}

Intervalle finden:

Ansteigend

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

Absteigend

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

Absteigend

: 0 < x < \frac{3}{5},

Ansteigend

: \frac{3}{5} < x < \infty

Stecke

x = - \frac{3}{5}

x^{5} - x^{3} + 1 : - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1)

Stecke

x = 0

x^{5} - x^{3} + 1 : 1

Sattel (0, 1)

Stecke

x = \frac{3}{5}

x^{5} - x^{3} + 1 : (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1

Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1), Sattel (0, 1), Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

e x t re m e y = \frac{x}{x ^{2} + 16}

e x t re m e - x^{3} (x - 3)

e x t re m e f (x) = \frac{4}{3} x^{3} - \frac{13}{2} x^{2} - 3 x + 11

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

e x t re m e f (x) = 8 x - 16 cos (x)