Answer :

Given,


the number 8 is divided into two numbers.


the product of the square of one number and cube of another number is minimum.


Let us consider,


x and y are the two numbers


Sum of the numbers : x + y = 8


Product of square of the one number and cube of anther number : S = x3 + y2


Now as,


x + y = 8


y = (8-x) ------ (1)


Consider,


S = x3 + y2


By substituting (1), we have


S = x3 + (8-x)2 ------ (2)


For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.


Differentiating the equation (2) with x





[Since ]



= 3x2 + 2x – 16 ------ (3)


Now equating the first derivative to zero will give the critical point c.


So,



Hence 3x2 + 2x - 16= 0






(or)


(or)


x = 2 (or) x = -2.67


Now considering the critical values of x = 2,-2.67


Now, for checking if the value of P is maximum or minimum at x=2,-2.67, we will perform the second differentiation and check the value of at the critical value x = 2,-2.67.


Performing the second differentiation on the equation (3) with respect to x.




= 3 (2x) + 2 (1) - 0


[Since and ]


= 6x + 2



Now when x = -2.67,



= - 16.02 + 2 = -14.02


At x = -2.67 hence, the function S will be maximum at this point.


Now consider x = 2,



= 12 + 2 = 14


Hence , so at x = 2, the function S is minimum


As second differential is positive, hence at the critical point x = 2 will be the maximum point of the function S.


Therefore, the function S is maximum at x = 2.


From Equation (1), if x= 2


y = 8 – 2 = 6


Therefore, x = 2 and y = 6 are the two positive numbers whose sum is 8 and the sum of the square of one number and cube of another number is maximum.


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