AP DSC School Assistant Math Model Test 3

Concept:We need to analyze the continuity and differentiability properties of each given function and match them with the correct descriptions. The greatest integer function $[x]$ is discontinuous at all integers, and absolute value functions are continuous but not differentiable at points where the expression inside becomes zero. Explanation:Consider each function one by one:(A) $|x-1| + |x-2|$This sum of absolute values is continuous everywhere. It has sharp corners at $x=1$ and $x=2$, so it is not differentiable at those points. Among the descriptions, "is continuous everywhere" (II) fits perfectly.(B) $x - |x|$For $x \ge 0$, $|x|=x$, so $x-x=0$; for $x < 0$, $|x|=-x$, so $x-(-x)=2x$. The function is continuous at $x=0$ because left and right limits equal $0$. However, the left-hand derivative is $2$ and the right-hand derivative is $0$, so it is not differentiable at $x=0$. It is differentiable everywhere else, matching description (I).(C) $x - [x]$This is the fractional part function ${\{x\}}$. It has a jump discontinuity at every integer. At $x=1$, the function is discontinuous, hence not differentiable. This matches description (III) "is not differentiable at $x=1$".(D) $x|x|$Write as $x \cdot |x|$. For $x \ge 0$, it equals $x^2$; for $x < 0$, it equals $-x^2$. The function is smooth everywhere, including at $x=1$. So it is differentiable at $x=1$, matching description (IV).Thus the correct matching is: (A) → (II), (B) → (I), (C) → (III), (D) → (IV).Answer:Option C: (A) - (II), (B) - (I), (C) - (III), (D) - (IV)