Question

已知冪函數(shù)f(x)=(t³-t+1)(t∈Z)是偶函數(shù)且在(0,+∞)上為增函數(shù)，求實(shí)數(shù)t的值.

Answer 1

∵f(x)是冪函數(shù),∴t³-t+1=1.∴t=-1,1或0.

當(dāng)t=0時(shí)f(x)=是奇函數(shù);當(dāng)t=-1時(shí)f(x)= 是偶函數(shù);

當(dāng)t=1時(shí)f(x)=是偶函數(shù);且,都大于0,在(0,+∞)上為增函數(shù).

故t=1且f(x)= 或t=-1且f(x)=.

解析:

關(guān)于冪函數(shù)y=xⁿ(n∈Q,n≠0)的奇偶性問(wèn)題,設(shè) (|p|、|q|互質(zhì)),當(dāng)q為偶數(shù)時(shí),p必為奇數(shù).y=xⁿ是非奇非偶函數(shù);當(dāng)q是奇數(shù)時(shí),y=xⁿ的奇偶性與p的奇偶性對(duì)應(yīng).