How many TVs will give maximum profit and what is the maximum profit?The daily profit, P, made by a TV manufacturer is given by P = 100x −...

The function for the profit earned per TV manufactured is given
as P = 100x − 0.02x^2. Now we need to find the number of TV's that should be manufactured to earn
the maximum profits. To solve this problem we need to differentiate the function P = 100x −
0.02x^2 with respect to x.

P’ = 100 –
2*0.02x

=> 100 – 0.04x

Now
equating P’ to zero:

=> 100 – 0.04x =
0

=> 100 = 0.04x

=> x =
100 / 0.04

=> x = 2500.

Now, to
determine if manufacturing these many TVs provides the maximum or minimum profit, we find P’’.
This is equal to -0.04 which is negative. Therefore x = 2500 provides the point of maximum
profit.

The maximum profit can be found by substituting x = 2500 in
P = 100 – 0.04x. We get 100*2500 – 0.02*2500 = 249,
950.

The required value of the number of TVs to be
manufactured to maximize profit is 2500. The maximum profit is $249,
950.