divide a number 10 into two parts such that product of square of one with cube of other is greatest
Answers
product of square of one with cube of other is greatest is 9 and 1
The two parts in which the number should be divided are 4 and 6.
Let the parts in which the number 10 is divided be x and (10-x) respectively.
Let f(x) be a function such that ,
f(x) = x³ × (10-x)²
It is given in the question that f(x) has maximum value.
Differentiating f(x) with respect to x
=> f'(x) = 3x²(10-x)² - 2x³(10-x)
=> f'(x) = x²(10-x) (3(10-x)-2x)
=> f'(x) = x²(10-x) (30-5x)
The function f(x) will have the maximum value for one of the points that satisfy the equation f'(x) = 0.
=> f'(x) = 0
=> x²(10-x) (30-5x) = 0
=> x = 0, 10, 6
Checking the value of f(x) for x = 0,10,6
At x = 0 , f(x) = 0
At x = 10 , f(x) = 0
At x = 6 , f(x) = 3456
The value of f(x) is maximum for x = 6.
The two parts of the number are 4 and 6.