【題目】某企業(yè)準(zhǔn)備投入適當(dāng)?shù)膹V告費(fèi)對(duì)產(chǎn)品進(jìn)行促銷，在一年內(nèi)預(yù)計(jì)銷售（萬(wàn)件）與廣告費(fèi)（萬(wàn)元）之間的函數(shù)關(guān)系為.已知生產(chǎn)此產(chǎn)品的年固定投入為3萬(wàn)元，每生產(chǎn)1萬(wàn)元此產(chǎn)品仍需再投入32萬(wàn)元，若每件銷售價(jià)為“平均每件生產(chǎn)成本的150%”與“年平均每件所占廣告費(fèi)的50%”之和.

（1）試將年利潤(rùn)（萬(wàn)元）表示為年廣告費(fèi)（萬(wàn)元）的函數(shù)；

（2）當(dāng)年廣告費(fèi)投入多少萬(wàn)元時(shí)，企業(yè)年利潤(rùn)最大？最大利潤(rùn)為多少？

【答案】（1）；（2）當(dāng)年廣告費(fèi)為萬(wàn)元時(shí)，企業(yè)利潤(rùn)最大，最大值為萬(wàn)元．

【解析】

試題分析：（1）由題意可得，產(chǎn)品的生產(chǎn)成本為萬(wàn)元，得到每萬(wàn)件銷售價(jià)，進(jìn)而得到年銷售輸入，即求解年利潤(rùn)的表達(dá)式；（2）令，則，利用基本不等式求解最值，即可得到結(jié)論.

試題解析：（1）由題意可得，產(chǎn)品的生產(chǎn)成本為萬(wàn)元，每萬(wàn)件銷售價(jià)為， （2分）

∴年銷售收入為， （4分）

∴年利潤(rùn)

. （6分）

（2）令，則

. （8分）

∵,∴,即， （10分）

當(dāng)且僅當(dāng)，即時(shí)，有最大值42，此時(shí).

即當(dāng)年廣告費(fèi)為7萬(wàn)元時(shí)，企業(yè)利潤(rùn)最大，最大值為42萬(wàn)元. （12分）

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