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Let f : R → R be defined as f(x) = { x⁵ sin(1/x) + 5x², x < 0 0, x = 0 x⁵ cos(1/x) + λx², x > 0 The value of λ for which f''(0) exists is ________.

1. 5
2. 5.00
3. 5 m
4. 5.0

Correct answer: 5.00

Solution

The value of λ must be chosen such that the second derivative of f at 0 exists, which requires the left-hand and right-hand limits of the second derivative to be equal at that point. In this case, both limits yield the same value when λ is set to 5, ensuring continuity and differentiability at x = 0.

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