Discussion
Let
\[ f(x) = \begin{cases} 1 - x, & x \lt 1 \\ (1 - x)(2 - x), & 1 \leq x \leq 2 \end{cases} \]
. Which one of the following is not true?
Answer: D
The statement that is not true is \(f(x)\) is not differentiable at \(x\) = 1.
At \(x = 1\) both pieces give 0, so \(f\) is continuous there. Compare the one-sided derivatives:
\[ \begin{aligned} \text{left: } \tfrac{d}{dx}(1 - x) &= -1 \\ \text{right: } \tfrac{d}{dx}(x^{2} - 3x + 2) &= 2x - 3 = -1 \end{aligned} \]
They are equal, so \(f\) is differentiable at 1, and the claim that it is not is false. The other statements are true: \(1 - x\) is differentiable everywhere left of 1 (including -2026 and the interval (-1, 1)), and at 1.5 the quadratic piece is smooth.
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