What is the inflection f(x)=x^2ln(x/8) ?

The inflection f(x)=x^2ln(x/8) is (8/(e^{3/2)},-(96)/(e^3))

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inflection f(x)=x^2ln(x/8)

inflection points

f (x) = x^{2} ln (\frac{x}{8})

(\frac{8}{e ^{\frac{3}{2}}}, - \frac{96}{e ^{3}})

Solution steps

f^{''} (x) = 2 ln (\frac{x}{8}) + 3

Find intervals:

Concave Downward

: 0 < x < \frac{8}{e ^{\frac{3}{2}}},

Concave Upward

: \frac{8}{e ^{\frac{3}{2}}} < x < \infty

Plug

x = \frac{8}{e ^{\frac{3}{2}}}

into

x^{2} ln (\frac{x}{8}) : - \frac{96}{e ^{3}}

(\frac{8}{e ^{\frac{3}{2}}}, - \frac{96}{e ^{3}})

f (x) = \frac{9 - x ^{2}}{2 x ^{2}}

x^{4} - x^{3} - 13 x^{2} + 25 x - 12

f (x) = x^{2} - 8 x + 4

f (x) = (9 - x^{2}) (x + 1)

(- 8, - 3), (2, 3)

What is the inflection f(x)=x^2ln(x/8) ?
The inflection f(x)=x^2ln(x/8) is (8/(e^{3/2)},-(96)/(e^3))