What is the extreme f(x)=ln(x+2)(x-1)^2 ?

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

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

extreme points

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)

Solution steps

Find the critical points:

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

Domain of

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

Find intervals:

Increasing

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

Decreasing

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

Increasing

: 1 < x < \infty

Plug the extreme point

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

into

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

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

Plug the extreme point

x = 1

into

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

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)