Divide 10 into two parts such that product of square of one with the cube of other may be greater?

Dear Student ,
 
Please find below the solution to the asked query :

Let the one number be xOther number be 10-xSo,P=x2 10-x3Differentiating wrt x on both sides , we getdPdx=x2 d 10-x3dx+10-x3 d x2dx=x2 -3 10-x2+10-x3 2x=-3 x2 10-x2+2 x 10-x3=x 10-x2 -3x+210-x=x 10-x2 -3x+20-2x=x 10-x2 -5x+20To find critical point dPdx=0x 10-x2 -5x+20=0x=0  or x=10 or x=4x=0 and x=10 not possible.So , x=4One part is 4 and other part is 6.d2Pdx2=10-x2 -5x+20-2x 10-x -5x+20-5x 10-x2 For x=4d2Pdx2=62 0 - 8 6 0-54 62=-720<0So, P is maximum at x=4
 
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