What is the inflection f(x)=5x^2ln(x/2) ?

The inflection f(x)=5x^2ln(x/2) is (2/(e^{3/2)},-(30)/(e^3))

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inflection f(x)=5x^2ln(x/2)

inflection points

f (x) = 5 x^{2} ln (\frac{x}{2})

(\frac{2}{e ^{\frac{3}{2}}}, - \frac{30}{e ^{3}})

Solution steps

f^{''} (x) = 5 (2 ln (\frac{x}{2}) + 3)

Find intervals:

Concave Downward

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

Concave Upward

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

Plug

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

into

5 x^{2} ln (\frac{x}{2}) : - \frac{30}{e ^{3}}

(\frac{2}{e ^{\frac{3}{2}}}, - \frac{30}{e ^{3}})

f (x) = \frac{( ln ( cos ( x ) ) )}{x}

x^{2} + 14 x + 43

f (x) = 3 x - 15

f (x) = \frac{x ^{2} + 2}{7 x - 2 x ^{2}}

\frac{x}{x ^{2} + 25}

What is the inflection f(x)=5x^2ln(x/2) ?
The inflection f(x)=5x^2ln(x/2) is (2/(e^{3/2)},-(30)/(e^3))