The function f(x) = |x| is differentiable at x = 0?

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Question: The function f(x) = |x| is differentiable at x = 0?

Options:

Correct Answer: No

Solution:

f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.

The function f(x) = |x| is differentiable at x = 0?

The function f(x) = |x| is differentiable at x = 0?

Step 1: Understand what differentiable means. A function is differentiable at a point if it has a defined slope (derivative) at that point.
Step 2: Look at the function f(x) = |x|. This function has two parts: when x is positive (f(x) = x) and when x is negative (f(x) = -x).
Step 3: Find the left-hand derivative at x = 0. This means looking at values of x that are slightly less than 0. The slope (derivative) from the left is -1.
Step 4: Find the right-hand derivative at x = 0. This means looking at values of x that are slightly more than 0. The slope (derivative) from the right is +1.
Step 5: Compare the left-hand and right-hand derivatives. The left-hand derivative is -1 and the right-hand derivative is +1.
Step 6: Since the left-hand and right-hand derivatives do not match (-1 ≠ +1), the function f(x) = |x| is not differentiable at x = 0.

Differentiability – The property of a function to have a derivative at a given point, which requires the function to be continuous and have matching left-hand and right-hand derivatives.
Absolute Value Function – The function f(x) = |x| is piecewise defined, leading to different behaviors on either side of x = 0.
Left-hand and Right-hand Derivatives – The derivatives calculated from the left and right sides of a point, which must be equal for differentiability.