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A function h(x) is defined as: h(x) = alpha * sqrt(x + 1), for 0 <= x <= 3 h(x) = beta * x + 2, for 3 < x <= 5 If h(x) is differentiable on (0, 5), find the value of {alpha + beta}, where {t} denotes the fractional part of t.

1. 0
2. 1/5
3. 1/3
4. None of these

Correct answer: None of these

Solution

Setting up continuity and differentiability conditions at x = 3 gives two equations: 2*alpha = 3*beta + 2 and alpha/4 = beta. Solving: alpha = 8/5, beta = 2/5. Then alpha + beta = 2 and {alpha + beta} = {2} = 0. Wait — re-checking: {2} = 0, so the answer is 0.

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