N characters of information are held on magnetic tape, in batches of x character each; the batch processing time is α+βx2 seconds, α, β are constants. The optimum value of x for fast processing is
√αβ
T=Nx(α+βx2)=N(αx+β )⇒ dTdx=N(−αx2+β)∴ dTdx=0⇒x2=αβ⇒ x=√αβAlso, d2Tdx2=N(2αx3)>0 at x=√αβ
∴ T is minimum for x=√αβ