Question

函數(shù)y=f′(x)是函數(shù)y=f(x)的導(dǎo)函數(shù),且函數(shù)y=f(x)在點(diǎn)P(x0,f(x0))處的切線為l:y=g(x)=f′(x0)·(x-x0)+f(x0),F(x)=f(x)-g(x), 如果函數(shù)y=f(x)在區(qū)間[a,b]上的圖象如圖所示,且a<x0<b,那么(　　)

(A)F'(x0)=0,x=x0是F(x)的極大值點(diǎn)

(B)F'(x0)=0,x=x0是F(x)的極小值點(diǎn)

(C)F'(x0)≠0,x=x0不是F(x)的極值點(diǎn)

(D)F'(x0)≠0,x=x0是F(x)的極值點(diǎn)

 

Answer 1

B

【解析】【思路點(diǎn)撥】y=g(x)是函數(shù)y=f(x)在點(diǎn)P(x0,f(x0))處的切線,故g'(x)= f'(x0),據(jù)此判斷F'(x0)是否為0,再進(jìn)一步判斷在x=x0兩側(cè)F'(x)的符號(hào).

解:F'(x)=f'(x)-g'(x)=f'(x)-f'(x0),

∴F'(x0)=f'(x0)-f'(x0)=0,又當(dāng)x<x0時(shí),從圖象上看,f'(x)<f'(x0),即F'(x)<0,此時(shí)函數(shù)F(x)=f(x)-g(x)為減函數(shù),同理,當(dāng)x>x0時(shí),函數(shù)F(x)為增函數(shù).