Determine if the function f(x) = |x - 1| is differentiable at x = 1.
Practice Questions
1 question
Q1
Determine if the function f(x) = |x - 1| is differentiable at x = 1.
Yes
No
Only from the left
Only from the right
The left-hand derivative is -1 and the right-hand derivative is 1. Since they are not equal, f(x) is not differentiable at x = 1.
Questions & Step-by-step Solutions
1 item
Q
Q: Determine if the function f(x) = |x - 1| is differentiable at x = 1.
Solution: The left-hand derivative is -1 and the right-hand derivative is 1. Since they are not equal, f(x) is not differentiable at x = 1.
Steps: 5
Step 1: Understand what it means for a function to be differentiable at a point. A function is differentiable at a point if the slope (derivative) from the left side and the right side at that point are the same.
Step 2: Identify the function we are working with, which is f(x) = |x - 1|.
Step 3: Find the left-hand derivative at x = 1. This means we look at values of x that are slightly less than 1 (like 0.9). For these values, f(x) = 1 - x, and the derivative is -1.
Step 4: Find the right-hand derivative at x = 1. This means we look at values of x that are slightly greater than 1 (like 1.1). For these values, f(x) = x - 1, and the derivative is 1.
Step 5: Compare the left-hand derivative (-1) and the right-hand derivative (1). Since -1 is not equal to 1, the function is not differentiable at x = 1.
