Need help with this please (calculus)?

Question

Suppose the total cost, C, (in dollars) to manufacture a quantity, x, (in hundreds of liters), is given by C=(x^3)/(3)-(33x^2)/(2)+272x+12290. Beyond what quantity does the cost continually increase without end?

I used parentheses to show which parts are included with what

Thanks for any help, I really need an answer

Umm it's supposed to be +12290 at the end im not quite sure why it got cut off

I'm not taking a test with this....my teacher gives us homework online and the program we use to do the homework asked this question but our teacher did not give us an example how to do it which is why i am clueless

Answer 1

The function is cubic, meaning that it will have a local maximum and a local minimum. Right after the local minimum, it will take off - i.e. increase without bound. So the question is asking you to find the location of that local minimum.

Take the first derivative and set it equal to 0. That will give you a quadratic, which will give you two solutions. One will be the maximum, one the minimum. Use either the first or the second derivative test (I'd recommend the first) to tell them apart, and you have your answer.

Answer 2

dC/dx = 3x^2 - 33x + 272

Setting dC/dx = 0 to find critical points, (inflection) and solving for "x":
x~ 4.7, 10.2

To find the "x" at which Cost increases without bound:

Per the Concavity Theorem:
f''(C) = 6x - 33
f''(4.7) < 0 (concave down, decreasing)
f''(10.2) > 0 (concave up, increasing)

SO: After the production of ~10.2 Liters, the production cost increases without bound.
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By the way,.. If you're taking a test right now and try to "use" this,.. you'll probably get busted,.. it'll be obvious!!!! SO DON'T,.. just try to understand the concepts so that you'll succeed in this course!

Answer 3

i dont know calculas but i do actually know how to do regular math i never done calculas before in my life but i might be able to help you