What is the extreme 8/x+2pix^2 ?

The extreme 8/x+2pix^2 is Minimum(\sqrt[3]{2/pi},4*2^{2/3}\sqrt[3]{pi}+2pi(2/pi)^{2/3})

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extreme 8/x+2pix^2

extreme

Minimum (3 \frac{2}{π}, 4 \cdot 2^{\frac{2}{3}} 3 π + 2 π (\frac{2}{π})^{\frac{2}{3}})

Solution steps

f^{'} (x) = - \frac{8}{x ^{2}} + 4 π x

Find intervals:

Decreasing

: - \infty < x < 0,

Decreasing

: 0 < x < \frac{2}{3 4 3 π},

Increasing

: \frac{2}{3 4 3 π} < x < \infty

Plug

x = 3 \frac{2}{π}

into

\frac{8}{x} + 2 π x^{2} : 4 \cdot 2^{\frac{2}{3}} 3 π + 2 π (\frac{2}{π})^{\frac{2}{3}}

Minimum (3 \frac{2}{π}, 4 \cdot 2^{\frac{2}{3}} 3 π + 2 π (\frac{2}{π})^{\frac{2}{3}})

What is the extreme 8/x+2pix^2 ?
The extreme 8/x+2pix^2 is Minimum(\sqrt[3]{2/pi},4*2^{2/3}\sqrt[3]{pi}+2pi(2/pi)^{2/3})