Question

Is the absolute value function differentiable at 0?
1) True
2) False

Answer 1

The absolute value function is not differentiable at 0 because it has a cusp at that point, resulting in different derivatives from the left and the right.

Step-by-step explanation:

The question asks whether the absolute value function is differentiable at 0. The answer is False. The absolute value function, which is defined as |x|, has a 'cusp' or a point at which the function changes direction at x=0. This makes the derivative at that point undefined because the slope of the function is not the same from the left and the right of the cusp. In essence, the function has a different left-hand derivative (which is -1) and a right-hand derivative (which is 1) at x=0, thus it is not differentiable at that point.