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

Lösung

extrempunkte

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

Maximum \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}, ln \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )} + 2 \frac{32 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2} - 36 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) + 9}{4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2}} + 1, Minimum (1, 0)

Schritte zur Lösung

Bestimme den kritischen Punkt:

x = \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}, x = 1

Bereich von

ln (x + 2) (x - 1)^{2} : x > - 2

Intervalle finden:

Ansteigend

: - 2 < x < \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )},

Absteigend

: \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )} < x < 1,

Ansteigend

: 1 < x < \infty

Setz die Extremwerte

x = \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}

ln (x + 2) (x - 1)^{2} \Rightarrow y = ln \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )} + 2 \frac{32 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2} - 36 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) + 9}{4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2}} + 1

ein

Maximum \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}, ln \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )} + 2 \frac{32 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2} - 36 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) + 9}{4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2}} + 1

Setz die Extremwerte

x = 1

ln (x + 2) (x - 1)^{2} \Rightarrow y = 0

ein

Minimum (1, 0)

Maximum \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}, ln \frac{3 - 4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )}{2 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} )} + 2 \frac{32 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2} - 36 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) + 9}{4 W _{0} ( \frac{3 e ^{\frac{1}{2}}}{2} ) ^{2}} + 1, Minimum (1, 0)

e x t re m e 2 x^{3} - 3 x^{2} - 12 x

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

e x t re m e f (x, y) = 4 x y - x^{2} y - x y^{2}

e x t re m e f (x) = x^{3} (x - 5)^{2}

e x t re m e f (x) = \frac{1}{4} x + 3 + \frac{400}{x}