Need help with a calculus question

united2k11

A t-shirt manufacturer is planning to expand its workforce. It estimates that the number of t-shirts produced by hiring a new workers is given by:
T(x) = -0.25^4 + 1x^3 0≤ x≤ 3

When is the rate of change of T-shirt production increasing, and when is it decreasing?
(the answer is (0,2) but i need to know how to do the workings for it)

What is the point of diminishing returns?
(the answer is x = 2 workers, again i need to know the workings for it)

What is the maximum rate of change for t-shirt production?
(answer is 4 ... need to know workings again)

Thanks for any help. I would appreciate it so much. I am horrible at these types of questions.

Yakuza

Are you sure about that function? Is the first term not missing an x somewhere? As written, it's[latex](\frac{-1}{4})^4 + x^3[/latex] which has no turning point in the range given.

Assuming you meant [latex]-(\frac{1}{4})x^4 + x^3[/latex], to find where the production starts to decrease, you need to find a point of inflection (http://en.wikipedia.org/wiki/Inflection_point) of that function. To do this, find the second derivative of the above function and set it to zero, then solve the equation. So, in the range of 0 to 3, you will have the turning point of where production starts to decrease.

united2k11

Yakuza wrote: »

Are you sure about that function? Is the first term not missing an x somewhere? As written, it's[latex](\frac{-1}{4})^4 + x^3[/latex] which has no turning point in the range given.

Assuming you meant [latex]-(\frac{1}{4})x^4 + x^3[/latex], to find where the production starts to decrease, you need to find a point of inflection (http://en.wikipedia.org/wiki/Inflection_point) of that function. To do this, find the second derivative of the above function and set it to zero, then solve the equation. So, in the range of 0 to 3, you will have the turning point of where production starts to decrease.

Thanks very much. And yes i forgot the x at -0.25x^4 . thanks!