6. Find the maximum profit that a company can make, if the profit function is
given by  p(x) = 41 - 72x - 18x ^2

Answers (1)

Profit of the company is given by the function 
p(x) = 41 - 72x - 18x ^2
p^{'}(x)= -72-36x\\ p^{'}(x) = 0\\ -72-36x= 0\\ x = -2
x = -2 is the only critical point of the function p(x) = 41 - 72x - 18x ^2
Now, by second derivative test
p^{''}(x)= -36< 0
At x = -2 p^{''}(x)< 0
Hence, maxima of function p(x) = 41 - 72x - 18x ^2 occurs at x = -2 and maximum value is
p(-2) = 41 - 72(-2) - 18(-2) ^2=41+144-72 = 113
Hence, the maximum profit the company can make is 113 units

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