What is the extreme f(x)=(24)/x+4pix^2 ?

The extreme f(x)=(24)/x+4pix^2 is Minimum(\sqrt[3]{3/pi},8*3^{2/3}\sqrt[3]{pi}+4pi(3/pi)^{2/3})

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extreme f(x)=(24)/x+4pix^2

extreme points

f (x) = \frac{24}{x} + 4 π x^{2}

Minimum (3 \frac{3}{π}, 8 \cdot 3^{\frac{2}{3}} 3 π + 4 π (\frac{3}{π})^{\frac{2}{3}})

Solution steps

f^{'} (x) = - \frac{24}{x ^{2}} + 8 π x

Find intervals:

Decreasing

: - \infty < x < 0,

Decreasing

: 0 < x < 3 \frac{3}{π},

Increasing

: 3 \frac{3}{π} < x < \infty

Plug

x = 3 \frac{3}{π}

into

\frac{24}{x} + 4 π x^{2} : 8 \cdot 3^{\frac{2}{3}} 3 π + 4 π (\frac{3}{π})^{\frac{2}{3}}

Minimum (3 \frac{3}{π}, 8 \cdot 3^{\frac{2}{3}} 3 π + 4 π (\frac{3}{π})^{\frac{2}{3}})

\frac{4 x}{x ^{2} + 1}, - 3 \leq x \leq 0

f (x) = \frac{x ^{3}}{3} - x^{2} - 8 x - 4

f (x) = x^{4} - 5 x^{3} + 1.1 x^{2} - 10.2 x + 1

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

f (x) = 2 csc (x)

What is the extreme f(x)=(24)/x+4pix^2 ?
The extreme f(x)=(24)/x+4pix^2 is Minimum(\sqrt[3]{3/pi},8*3^{2/3}\sqrt[3]{pi}+4pi(3/pi)^{2/3})