Finding the minimum cost from the given question. Please help.

Question

) A company is hiring persons to work in its plant. For the person to perform the job efficiently,

experts estimate that the cost C, of performing the task is a function of the number of persons

hired specifically, C(x)= 0.005x^2 - 49 lnx + 5

i) Determine the number of persons who should be hired to minimize the cost

ii) what is the minimum cost

Answer 1

We are given a cost function C(x) = 0.005x² - 49ln(x) + 5

To find the minimum (or maximum) of a function we need to differentiate:

^d/_dx [ 0.005x² - 49ln(x) + 5 ]

Which is the same as ^d/_dx [ 0.005x² ] - ^d/_dx [ 49ln(x) ] + ^d/_dx [ 5 ]

=> 0.01x - 49 (¹/_x) + 0

AND at a minimum (or maximum) we know this differential has a value of 0 (because at any "turning point" the tangent to the function is horizontal, and has a gradient of zero).

Hence, 0.01x - 49 (¹/_x) = 0

=> ^x/₁₀₀ = ⁴⁹/_x

=> x² = 4900

=> x = √(4900)

=> x = ±70

We can't have negative persons, so (i) minimum cost is at 70 persons.

To find cost, substitute this value in the initial cost function, giving

cost = 0.005(70)² - 49ln(70) + 5

=> 24.5 - 208.1763 + 5 = -178.68

so (ii) the minimum cost is -178.68 (I presume $, but question doesn't say, and while a negative cost is unusual, it is possible).

You can use Demos Graphing or similar to see what this function looks like, and where the minimum occurs.

https://www.desmos.com/calculator/h3dsgqtnhp