Question

已知函數(shù)f(x)=x³-x²+x+b,其中a,b∈R.
(1)若曲線y=f(x)在點(diǎn)P(2,f(2))處的切線方程為y=5x-4,求函數(shù)f(x)的解析式.
(2)當(dāng)a>0時(shí),討論函數(shù)f(x)的單調(diào)性.

Answer 1

(1) f(x)=x³-2x²+x+4
(2) 當(dāng)0<a<1時(shí),>1,函數(shù)f(x)在區(qū)間(-∞,1)及(,+∞)上為增函數(shù),在區(qū)間(1,)上為減函數(shù);
當(dāng)a=1時(shí),=1,函數(shù)f(x)在區(qū)間(-∞,+∞)上為增函數(shù);
當(dāng)a>1時(shí),<1,函數(shù)f(x)在區(qū)間(-∞,)及(1,+∞)上為增函數(shù),在區(qū)間(,1)上為減函數(shù).

(1)f'(x)=ax²-(a+1)x+1.
由導(dǎo)數(shù)的幾何意義得f'(2)=5,于是a=3.
由切點(diǎn)P(2,f(2))在直線y=5x-4上可知2+b=6,解得b=4.
所以函數(shù)f(x)的解析式為f(x)=x³-2x²+x+4.
(2)f'(x)=ax²-(a+1)x+1=a(x-)(x-1).
當(dāng)0<a<1時(shí),>1,函數(shù)f(x)在區(qū)間(-∞,1)及(,+∞)上為增函數(shù),在區(qū)間(1,)上為減函數(shù);
當(dāng)a=1時(shí),=1,函數(shù)f(x)在區(qū)間(-∞,+∞)上為增函數(shù);
當(dāng)a>1時(shí),<1,函數(shù)f(x)在區(qū)間(-∞,)及(1,+∞)上為增函數(shù),在區(qū)間(,1)上為減函數(shù).