Consider the function 𝑓:ℝ→ℝ defined as follows:
where 𝑐1 ,𝑐2∈ℝ.
If 𝑓 is continuous at 𝑥 = 0, then 𝑐1+𝑐2 = _________. (answer in integer)

Question

Consider the function 𝑓:ℝ→ℝ defined as follows: where 𝑐1 ,𝑐2∈ℝ. If 𝑓 is continuous at 𝑥 = 0, then 𝑐1+𝑐2 = _________. (answer in integer)

Accepted Answer

Correct answer: 3 — For x ≤ 0, f(x) = 3, so f(0) = 3 and the left-hand limit at 0 is 3. For x > 0, f(x) = c₁eˣ - c₂ ln(1/x). As x → 0+, ln(1/x) → ∞. For the right-hand limit to be finite, we must have c₂ = 0. Then the right-hand limit becomes c₁e⁰ = c₁. Continuity at x = 0 requires c₁ = 3. Hence c₁ + c₂ = 3 + 0 = 3.

Consider the function 𝑓:ℝ→ℝ defined as follows: where 𝑐1 ,𝑐2∈ℝ. If 𝑓 is…