已知函數(shù)f(x)=4x3+3tx2-6t2x+t-1,x∈R,其中t∈R.
(1)當(dāng)t=1時,求曲線y=f(x)在點(0,f(0))處的切線方程;
(2)當(dāng)t≠0時,求f(x)的單調(diào)區(qū)間.
解:(1))當(dāng)t=1時,f(x)=4x
3+3x
2-6x,f(0)=0,f'(x)=12x
2+6x-6(2分)f'(0)=-6.所以曲線y=f(x)在點(0,f(0))處的切線方程為y=-6x.(4分)
(2)解:f'(x)=12x
2+6tx-6t
2,令f'(x)=0,解得x=-t或

.(5分)
因為t≠0,以下分兩種情況討論:
(i)若t<0,則t<0,則

,當(dāng)x變化時,f'(x),f(x)的變化情況如下表:
x |  |  | (-t,+∞) |
f'(x) | + | - | + |
f(x) | ↑ | ↓ | ↑ |
所以,f(x)的單調(diào)遞增區(qū)間是

的單調(diào)遞減區(qū)間是

. (8分)
(ii)若
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,當(dāng)x變化時,f'(x),f(x)的變化情況如下表:
x | (-∞,t) |  |  |
f'(x) | + | - | + |
f(x) | ↑ | ↓ | ↑ |
所以,f(x)的單調(diào)遞增區(qū)間是

的單調(diào)遞減區(qū)間是

.(12分)
分析:(1)當(dāng)t=1時,求出函數(shù)f(x),利用導(dǎo)數(shù)的幾何意義求出x=0處的切線的斜率,利用點斜式求出切線方程;
(2)根據(jù)f'(0)=0,解得x=-t或x=

,討論t的正負,在函數(shù)的定義域內(nèi)解不等式fˊ(x)>0和fˊ(x)<0求出單調(diào)區(qū)間即可.
點評:本題主要考查了導(dǎo)數(shù)的幾何意義,利用導(dǎo)數(shù)研究函數(shù)的單調(diào)性、曲線的切線方程、函數(shù)零點、解不等式等基礎(chǔ)知識,考查了計算能力和分類討論的思想.