The given polynomial is
[tex]p(x)=-0.002x^2+2.5x-600[/tex]
Differentiate with respect to x, we get
[tex]p^{\prime}(x)=2(-0.002)x+2.5[/tex]
[tex]p^{\prime}(x)=-0.002x+2.5[/tex]
Equate this to zero, we get
[tex]-0.004x+2.5=0[/tex]
[tex]-0.004x=-2.5[/tex]
[tex]0.004x=2.5[/tex]
multiply by 1000 on both sides, we get
[tex]0.004x\times1000=2.5\times1000[/tex]
[tex]4x=2500[/tex]
[tex]x=\frac{2500}{4}=625[/tex]
Hence they must sell 625 patterns to attain maximum profit.
Substitute x=625 in the given equation p(x), we get
[tex]P(625)=-0.002(625)^2+2.5(625)-600[/tex]
[tex]=-0.002\times390625+2.5\times625-600[/tex]
[tex]=-781.25+1562.5-600[/tex][tex]P(625)=181.25[/tex]
Hence the maximum profit is $ 181.25.
