Question

If Mr. Rane order x chairs at the price p = (2x² - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?

Solution: Let Mr. Rane order x chairs.

Then the total price of x chairs = p·x = (2x² - 12x- 192)x

= 2x³ - 12x² - 192x

Let f(x) = 2x³ - 12x² - 192x

∴ f'(x) = `square` and f''(x) = `square`

f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0

∴ f is minimum when x = 8

Hence, Mr. Rane should order 8 chairs for minimum cost of deal.

Answer 1

Let Mr. Rane order x chairs.

Then the total price of x chairs = p·x = (2x² - 12x- 192)x

= 2x³ - 12x² - 192x

Let f(x) = 2x³ - 12x² - 192x

`= 2xx3x^2-12xx2x-192xx1`

∴ f'(x) = 6x² - 24x - 192

and f''(x) = `d/dx(6x^2-24x-192)`

`=6xx2x-24xx1-0`

f''(x) = 12x - 24

f'(x ) = 0 gives 6x² - 24x - 192 = 0

∴ x² - 4x - 32 = 0

∴ (x - 8)(x + 4) = 0

∴ x - 8 = 0     ...[∵ x > 0]

∴ x = 8

and f''(8) = 12(8) - 24 = 96 - 24

= 72 > 0

∴ f is minimum when x = 8

Hence, Mr. Rane should order 8 chairs for minimum cost of deal.