Question

Complete the following activity to divide 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 is 84 - x

Letf (x) = x² (84 - x) = 84x² - x³

∴ f'(x) = `square`

and f''(x) = `square`

For extreme values, f'(x) = 0

∴ x = `square  "or"    square`

f(x) attains maximum at x = `square`

Hence, the two parts of 84 are 56 and 28.

Answer 1

Let one part be x. Then the other part is 84 - x

Letf (x) = x² (84 - x) = 84x² - x³

∴ f'(x) = `bb(168x - 3x^2)`

and f''(x) = `bb(168-6x)`

For extreme values, f'(x) = 0

∴ x = 0 or 56

Now,  f''(0) = 168 - 6(0) = 168 < 0

∴ f (x) attain minimum at x = 0

Also, f''(56) = 168 - 6(56) 

= 168 - 336 = -168 < 0

f(x) attains maximum at x = 56

Hence, the two parts of 84 are 56 and 28.