1. The problem asks whether the derivative of the function $f(x) = |x|$ is defined and equals 0 at $x=0$. 2. Recall the definition of the derivative at a point $x=a$: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ 3. For $f(x) = |x|$, we evaluate the left-hand and right-hand limits at $x=0$: - Left-hand limit ($h \to 0^-$): $$\lim_{h \to 0^-} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$$ - Right-hand limit ($h \to 0^+$): $$\lim_{h \to 0^+} \frac{|0+h| - |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$$ 4. Since the left-hand and right-hand limits are not equal, the derivative at $x=0$ does not exist. 5. Therefore, the statement "The derivative of $f(x) = |x|$ is defined and equals 0 at $x=0$" is **False**. Final answer: False