Question

Divide the number 84 into two parts such that the product of one part and square of the other is maximum. 

Solution:

Let one part be x then the other part will be 84 − x.

∴ f(x) = ${\displaystyle \square }$

∴ f'(x) = 168x − 3x²

For extreme values f'(x) = 0

168x − 3x² = 0

∴ 3x(56 − x) = 0

∴ x = ${\displaystyle \square \text{or}\square }$

f'(x) = 168 − 6x

If x = 0, f'(0) = 168 − 6(0) = 168 > 0

∴ function attains maximum at x = 0

If x = 56, f'(56) = ${\displaystyle \square }$ < 0

∴ function attains maximum at x = 56

∴ Two parts of 84 are ${\displaystyle \square \text{and}\square }$

Answer 1

Let one part be x then the other part will be 84 − x.

∴ f(x) = x(84 − x)²

∴ f'(x) = 168x − 3x²

For extreme values f'(x) = 0

168x − 3x² = 0

∴ 3x(56 − x) = 0

∴ x = 0 or 56

f'(x) = 168 − 6x

If x = 0, f'(0) = 168 − 6(0) = 168 > 0

∴ function attains maximum at x = 0

If x = 56, f'(56) = −168 < 0

∴ function attains maximum at x = 56

∴ Two parts of 84 are 56 and 28