[填空题]
The cost to manufacture badminton rackets at a company in J7ch5c q;+,tlsvsblm m 130u qyapan can pa l2 3afa2j4ybe modelled by t3a faj pa24y2lhe cost function
$C(x)=8 x^{3}-24 x^{2}+28x$
where x is in hundreds of rackets and C(x) is in hundreds of Japanese Yen (JPY).
1. Find $C^{\prime}(x)$ = a$x^2$ - bx + c;a= ,b= ,c= .
The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.
2. Find the marginal cost when 100 rackets are produced and interpret its meaning in this context.
$C^{\prime}(1)$=JPY .
The revenue from selling the rackets is given by the function R(x)=26 x where x is in hundreds of rackets and R(x) is in hundreds of JPY.
3. Given that Profit = Revenue - Cost, determine a function for the profit, P(x) , in hundreds of JPY from selling x hundreds of badminton rackets.
P(x)=-a$x^3$+24x$^b$-cx ; a= ,b= ,c= .
4. Find $P^{\prime}(x)$= -ax$^2$+bx-c; a= ,b= ,c= ..
5. Determine the intervals where P(x) is increasing and decreasing.
The derivative $P^{\prime}(x)$ gives the marginal profit. The production will reach its optimal level when the marginal profit is zero and P(x) is positive.
6. Find the optimal production level and the expected profit at this level.
