Zylinder:
HB:
O(r,h)=2r^2*π+2*r*π*h soll minimal werden
NB: V=r^2π*h → h=\( \frac{V}{r^2π} \)
O(r)=2r^2*π+2*r*π*\( \frac{V}{r^2π} \)=2r^2*π+\( \frac{2V}{r} \)= \( \frac{2 r^3*π+2V}{r} \)
\( O^{\prime}(r)=\frac{6 \cdot r^{2} \cdot \pi \cdot r-\left(2 r^{3} \pi+2 V\right) \cdot 1}{r^{2}}=\frac{4 r^{3} \cdot \pi-2 V}{r^{2}} \)
\( \left(4 r^{3} \cdot \pi-2 V\right)=0 \)
\( 2 r^{3} \cdot \pi=V \)
\( r=\sqrt[3]{\frac{V}{2 \pi}} \)
\( h=\ldots \)