A T-shirt manufacturer is planning to expend its workforce. It estimates that th

Question

A T-shirt manufacturer is planning to expend its workforce. It estimates that the number of T-shirts produced by hiring x new workers is given by T(x) = - 0.25x4 + 2x3 o x 6 When is the rate of change of TT-shirt production increasing and when is it decreasing? What is the point of diminishing returns and the maximum rate of change for T-shirt production? Graph T and T' on the same coordinate system. The rate of change of T-shirt production is increasing on (0,4). The rate of change of T-shirt production is decreasing on .

Explanation / Answer

T'(x) = -x^3 + 6x^2

=> x^3 - 6x^2 = 0

=> (x^2)(x - 6) = 0

x = 0 or x = 6 are the critical points.

If x = -1, T'(-1) = 1 + 6 = 7 > 0

If x = 1, T'(1) = -1 + 6 = 7 > 0

If x = 7, T'(1) = -243 + 6(49) < 0

So it is increasing on (-infinty, 6)

It is decreasing on (6, infinity)

But since the domain is restricted,

It is increasing on (0,6) and decreasing nowhere.

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