The cost function for production of a commodity is C(x)=339+25x-0.09x^{2}+0.0004x^{3} Find and

The cost function for production of a commodity is
$C\left(x\right)=339+25x-0.09{x}^{2}+0.0004{x}^{3}$
Find and interpret ${C}^{\prime }\left(100\right)$
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Step 1
Find ${C}^{\prime }$
$C\left(x\right)=339+25x-0.09{x}^{2}+0.0004{x}^{3}$
${C}^{\prime }\left(x\right)=25-0.18x+0.0012{x}^{2}$
Plug in $x=100$
${C}^{\prime }\left(100\right)=25-0.18\left(100\right)+0.0012{\left(100\right)}^{2}=19$
Step 2
${C}^{\prime }\left(100\right)=19$, the approximate cost increase of producing the 101st item when you produce 100 items
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Step 1
$C\left(x\right)=0.0004{x}^{3}-0.09{x}^{2}+25x+339$
${C}^{\prime }\left(x\right)=0.0012{x}^{2}-0.18x+25$
${C}^{\prime }\left(100\right)=12-18+25=19$
Marginal cost at 100 units is $\mathrm{}19$ per unit.