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Beliebt Funktionen >

extreme f(x)=x^2(x-2)^2(x-1)^2

Lösung

extrempunkte

f (x) = x^{2} (x - 2)^{2} (x - 1)^{2}

Lösung

Minimum (0, 0), Maximum (\frac{3 - 3}{3}, \frac{4}{27}), Minimum (1, 0), Maximum (\frac{3 + 3}{3}, \frac{4}{27}), Minimum (2, 0)

Schritte zur Lösung

f^{'} (x) = 2 x (x - 2) (x - 1) (3 x^{2} - 6 x + 2)

Intervalle finden:

Absteigend

: - \infty < x < 0,

Ansteigend

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

Absteigend

: \frac{3 - 1}{3} < x < 1,

Ansteigend

: 1 < x < \frac{1 + 3}{3},

Absteigend

: \frac{1 + 3}{3} < x < 2,

Ansteigend

: 2 < x < \infty

Stecke

x = 0

in

x^{2} (x - 2)^{2} (x - 1)^{2} : 0

Minimum (0, 0)

Stecke

x = \frac{3 - 3}{3}

in

x^{2} (x - 2)^{2} (x - 1)^{2} : \frac{4}{27}

Maximum (\frac{3 - 3}{3}, \frac{4}{27})

Stecke

x = 1

in

x^{2} (x - 2)^{2} (x - 1)^{2} : 0

Minimum (1, 0)

Stecke

x = \frac{3 + 3}{3}

in

x^{2} (x - 2)^{2} (x - 1)^{2} : \frac{4}{27}

Maximum (\frac{3 + 3}{3}, \frac{4}{27})

Stecke

x = 2

in

x^{2} (x - 2)^{2} (x - 1)^{2} : 0

Minimum (2, 0)

Minimum (0, 0), Maximum (\frac{3 - 3}{3}, \frac{4}{27}), Minimum (1, 0), Maximum (\frac{3 + 3}{3}, \frac{4}{27}), Minimum (2, 0)

Graph

Beliebte Beispiele

asymptoten f(x)=((x^2+1))/(x^3+1)

a sy m pt o t es f (x) = \frac{( x ^{2} + 1 )}{x ^{3} + 1}

invers f(x)= 1/27 x^3

in v erse f (x) = \frac{1}{27} x^{3}

domäne f(x)=sqrt(x^2-2x-8)

d o main f (x) = x^{2} - 2 x - 8

verschiebung 4/5 sin(-1/3 x)

s hi f t \frac{4}{5} sin (- \frac{1}{3} x)

domäne f(x)=sqrt(x^4-256)

d o main f (x) = x^{4} - 256