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 − 0.02x^2, where x is the number of TV's sold.
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.