Mathematics

The key is to recognize that $|x|$ creates a potential break in smoothness at $x = 0$. Rewrite the function by considering what happens on each side of zero: when $x \geq 0$, the absolute values simplify one way, and when $x < 0$, they simplify differently. Once you have the piecewise form, check whether the function itself is continuous at $x = 0$, and then whether the left and right derivatives match there. A function can only be differentiable where it's both continuous and has matching one-sided derivatives.

The key is to recognize that $|x|$ creates a potential break in smoothness at $x = 0$. Rewrite the function by considering what happens on each side of zero: when $x \geq 0$, the absolute values simplify one way, and when $x < 0$, they simplify differently. Once you have the piecewise form, check whether the function itself is continuous at $x = 0$, and then whether the left and right derivatives match there. A function can only be differentiable where it's both continuous and has matching one-sided derivatives.