Determine if the function f(x) = |x - 1| is differentiable at x = 1.
Practice Questions
Q1
Determine if the function f(x) = |x - 1| is differentiable at x = 1.
Yes
No
Only from the left
Only from the right
Questions & Step-by-Step Solutions
Determine if the function f(x) = |x - 1| is differentiable at x = 1.
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.
Differentiability – The property of a function to have a derivative at a given point, which requires the left-hand and right-hand derivatives to be equal.
Absolute Value Functions – Functions that can have sharp corners or cusps, which often lead to points of non-differentiability.
Left-hand and Right-hand Derivatives – The derivatives calculated from the left and right sides of a point, used to determine differentiability.
