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【題目】某企業(yè)準備投入適當?shù)膹V告費對產品進行促銷，在一年內預計銷售（萬件）與廣告費（萬元）之間的函數(shù)關系為.已知生產此產品的年固定投入為3萬元，每生產1萬元此產品仍需再投入32萬元，若每件銷售價為“平均每件生產成本的150%”與“年平均每件所占廣告費的50%”之和.

（1）試將年利潤（萬元）表示為年廣告費（萬元）的函數(shù)；

（2）當年廣告費投入多少萬元時，企業(yè)年利潤最大？最大利潤為多少？

【答案】（1）；（2）當年廣告費為萬元時，企業(yè)利潤最大，最大值為萬元．

【解析】

試題分析：（1）由題意可得，產品的生產成本為萬元，得到每萬件銷售價，進而得到年銷售輸入，即求解年利潤的表達式；（2）令，則，利用基本不等式求解最值，即可得到結論.

試題解析：（1）由題意可得，產品的生產成本為萬元，每萬件銷售價為，（2分）

∴年銷售收入為，（4分）

∴年利潤

. （6分）

（2）令，則

. （8分）

∵,∴,即，（10分）

當且僅當，即時，有最大值42，此時.

即當年廣告費為7萬元時，企業(yè)利潤最大，最大值為42萬元. （12分）

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