Question

Calculus help please.
find the maximum profit given the following revenue and cost functions
R(x) =104x - x^2; C(x) = 1/3x^3 -3x^2 + 92x+37

Answer 1

Hmm I'm not familiar with this business stuff...
Profit is Revenue minus Costs? Is that right?\[\large\rm P(x)=R(x)-C(x)\]

Answer 2

The formula is correct and the answer is supposed to be 35, but I don't see how they are arriving at that answer

Answer 3

■ \(\color{black}{\rm Step~\text{#}1}\)
Find the profit function: \(\color{blue}{\rm P(x)=R(x)-C(x)}\)


■ \(\color{black}{\rm Step~\text{#}2}\)
Maximize \(\color{blue}{\rm P(x)}\)

\(\color{black}{\rm Steps~~maximizing~~P'(x),~~are~~as~~follows:}\)
─ Find P\('\)(x)
─ Find all critical numbers.
─ Evaluate P(x), at each critical-number, and the maximum output
will be the maximum profit that you need . (The particular critical
number at which you obtain the maximum output for P(x), is the
price at which you obtained this maximum profit).

Answer 4

Additional note for finding the \(\LARGE \color{blue}{_{^{^{\bf critical~numbers}}}}\) for any function F(x).

At first, you have to find f\('\)(x).
Then you have 3 possible cases of critical numbers.


\(\bullet\) \(\color{black}{\rm Case~\text{#}1}\)
A critical number is any value of x, at which f\('\)(x) is undefined, PROVIDED
that this value of x is in the domain of the f(x). \(\color{red}{\small (}\)So, if f(x) is undefined
at this x-value, or if this x-value is not within the given interval of the f(x),
THEN this value of x is NOT a critical number.\(\color{red}{\small )}\)



\(\bullet\) \(\color{black}{\rm Case~\text{#}2}\)
We set the derivative of the function to be equivalent to 0, i.e. f\('\)(x)=0
and any x-solution to the equation f\('\)(x)=0 is a critical number. \(\color{red}{\small (}\)PROVIDED
that the x-solution for the equation f\('\)(x)=0 is within the domain of the f(x),
and if not, then this x-value is not a critical number.\(\color{red}{\small )}\)



\(\bullet\) \(\color{black}{\rm Case~\text{#}3}\)
Any closed-interval boundary of f(x) is a critical number.

\(\color{red}{\small (}\)So, if the interval of f(x) is [a,b] then x=a and x=b are critical numbers.
If the interval of f(x) is (a,b] then only x=b is a critical number.
If the interval of f(x) is [a,b) then only x=a is a critical number.
If the interval of f(x) is (a,b) then none of a and b are critical number.\(\color{red}{\small )}\)