Max-Min
A consulting firm conducts training sessions for employees of various companies. The charge to a company sending employees to a session is $50 per employee, less $0.50 for each employee in excess of 10. That is, for example, if twelve employees are sent, the charge per employee would be $49.00 and the total prorated charge to the company would be 12(49.00)=$558.00. The consulting firm further has a fixed total charge for groups of x or more where x is the number that maximizes the prorated group charge. What should x be, and what is the maximum total group charge to the company?

Question

Max-Min
A consulting firm conducts training sessions for employees of various companies. The charge to a company sending employees to a session is $50 per employee, less $0.50 for each employee in excess of 10. That is, for example, if twelve employees are sent, the charge per employee would be $49.00 and the total prorated charge to the company would be 12(49.00)=$558.00. The consulting firm further has a fixed total charge for groups of x or more where x is the number that maximizes the prorated group charge. What should x be, and what is the maximum total group charge to the company?

anonymous · Answer

Have you formed any equations?

anonymous · Answer

c={50 if 10

anonymous · Answer

You don't have to worry about the case x<= 10 because the example in the problem shows that revenue is greater at x=12 than at x=10. You have a good formula, just distribute x, take the derivative, set it equal to zero, and solve for x. Then check to make sure your result is a maximum.

anonymous · Answer

i get x=55 and maximum 1512.5 
can u please explain this to step by step i really need to understand this

anonymous · Answer

The derivative tells the slope of a curve. At a maximum or minimum point, the slope has to be zero. Therefore, you can locate potential maximum and minimum values by finding the derivative, setting it equal to zero, and solving the equation. Keep in mind that this process merely tells you where the slope is zero, and you always have to check whether it's a max or a min (or sometimes neither). This problem was easy in that regard because the equation was a quadratic with the highest term negative, so it's obvious that it's a parabola that opens down, and the place where the slope is zero has to be the max.

ranga · Answer

You have already solved the problem.  

The consulting firm further has a fixed total charge for groups of x or more ****where x is the number that maximizes the prorated group charge****. 
What should x be, and what is the maximum total group charge to the company?

You already found x = 55 maximizes the prorated group charge.
And the group rate is $1512.50

anonymous · Answer

thank u @ranga  and @creeksider so much for explaining

ranga · Answer

You are welcome.