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Find the value of k for which the function f(x) = x² - 2*(k+1)*x + 2 for x >= 1 and f(x) = x - 1 for x < 1 is both continuous and differentiable everywhere.

1. 0
2. -1/2
3. 1/2
4. Does not exist

Correct answer: Does not exist

Solution

Continuity at x=1: left limit f(1⁻) = 1-1 = 0; right value f(1) = 1 - 2(k+1) + 2 = 3 - 2k. Setting equal: 3 - 2k = 0 -> k = 3/2. Differentiability at x=1: left derivative = d/dx(x-1)|ₓ₌₁ = 1; right derivative = d/dx(x² - 2(k+1)x + 2)|ₓ₌₁ = 2 - 2(k+1) = -2k. For k=3/2: right derivative = -3 ≠ 1. The two conditions cannot be satisfied simultaneously by any single k, so the answer is 'Does not exist'.

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