Question:

Step 1: Analyze the condition $a^2 + b^2 + c^2 + ab + bc + ca \leq 0$. Multiply by 2: $$2a^2 + 2b^2 + 2c^2 + 2ab + 2bc + 2ca \leq 0$$ $$(a+b)^2 + (b+c)^2 + (c+a)^2 \leq 0$$ Since each square is $\geq 0$, we must have $a+b = 0$, $b+c = 0$, $c+a = 0$, giving $a = b = c = 0$. Step 2: Substitute $a = b = c = 0$ into $f(x)$: $$f(x) = 0 \cdot [x] + 0 \cdot |x| + 0 \cdot \text{sgn}(x) = 0$$ Step 3: Since $f(x) = 0$ for all $x \in (-2,2)$, $f(x)$ is continuous and derivable everywhere in $(-2,2)$. Step 4: Evaluate each option: - Option (a): $f(x)$ discontinuous at exactly two points — FALSE (f is identically 0, continuous everywhere) - Option (b): $f(x)$ discontinuous at exactly three points — FALSE - Option (c): $f(x)$ is continuous and derivable for every $x$ — TRUE - Option (d): $f(x)$ is non-derivable at exactly one point — FALSE Step 5: The options that are NOT true are (a), (b), and (d), i.e., options 1, 2, and 4.