Determine the point at which the function f(x) = |x - 1| is not differentiable.
Practice Questions
Q1
Determine the point at which the function f(x) = |x - 1| is not differentiable.
x = 0
x = 1
x = 2
x = -1
Questions & Step-by-Step Solutions
Determine the point at which the function f(x) = |x - 1| is not differentiable.
Correct Answer: x = 1
Step 1: Understand what the function f(x) = |x - 1| means. This function represents the distance between x and 1 on a number line.
Step 2: Identify where the function changes its behavior. The absolute value function has a 'corner' or 'cusp' where the expression inside the absolute value equals zero.
Step 3: Set the inside of the absolute value to zero: x - 1 = 0. Solve for x. This gives x = 1.
Step 4: Recognize that at x = 1, the function has a sharp point (cusp) and does not have a defined slope (derivative) at that point.
Step 5: Conclude that the function f(x) = |x - 1| is not differentiable at x = 1.
Absolute Value Function – The function |x - 1| represents the distance from x to 1, which creates a V-shape graph with a cusp at x = 1.
Differentiability – A function is differentiable at a point if it has a defined slope (derivative) at that point; cusps or corners indicate non-differentiability.
