Prove that the function f(x)=|x−6| is not differentiable at 6. Find formula for f′ and sketch its graph.
From the definition of absolute value,
f(x)=|x−6|={x−6ifx>0−(x−6)ifx<0⟹f(x)={x−6ifx>0−x+6ifx<0
From the definition of derivative,
f′+(x)=limh→0+f(x+h)−f(x)h and f′−(x)=limh→0−f(x+h)−f(x)h
For Right Hand,
f′+(x)=limh→0+x+h−6−(x−6)hf′+(x)=limh→0+\cancel−6−\cancelx+\cancel6hf′+(x)=limh→0+\cancelh\cancelhf′+(x)=limh→0+1f′+(6)=1
For Left Hand,
f′−(x)=limh→0−−(x+h)+6−(−x+6)hf′−(x)=limh→0−\cancel−x−h+\cancel6+\cancelx−\cancel6hf′−(x)=limh→0−\cancelh\cancelhf′−(x)=limh→0−−1f′−(6)=−1
f′+(6)≠f′−(6); Therefore, f is not differentiable at 6.
We can find the formula for f′ using f′−(x)=−1 and f′(x)=1
Therefore, we can rewrite f′(x) as
f′(x)={1ifx≥6−1ifx<6
No comments:
Post a Comment