[填空题]
The cost to manufactub(9w0n9csem vre badminton rackets at-uh f lmc0(y7 d+bdam )xr.t1ir v8)wv a company in Japan can be modelled by thafrudymi)r0.7d)lx8vw(-+c vm 1 hb te 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.
