Question

函數y=f′(x)是函數y=f(x)的導函數,且函數y=f(x)在點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y=f(x)在區間[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

B．F'(x0)=0,x=x0是F(x)的極小值點

C．F'(x0)≠0,x=x0不是F(x)的極值點

D．F'(x0)≠0,x=x0是F(x)的極值點

Answer 1

B

【思路點撥】y=g(x)是函數y=f(x)在點P(x₀,f(x₀))處的切線,故g'(x)= f'(x₀),據此判斷F'(x₀)是否為0,再進一步判斷在x=x₀兩側F'(x)的符號.
解:F'(x)=f'(x)-g'(x)=f'(x)-f'(x₀),
∴F'(x₀)=f'(x₀)-f'(x₀)=0,又當x<x₀時,從圖象上看,f'(x)<f'(x₀),即F'(x)<0,此時函數F(x)=f(x)-g(x)為減函數,同理,當x>x₀時,函數F(x)為增函數.