Let [x] denote the greatest integer less than or equal to x . If the function f(x) = (x^2 - ax + b)[x] is continuous at exactly two integers x = 2 and x = 3 , where a and b are real constants, then the value of a^2 + b^2 is _____.

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Correct answer: 61

For the function f(x) = (x^2 - ax + b)[x] to be continuous at an integer k , the left-hand limit (LHL) and right-hand limit (RHL) at x = k must be equal. LHL at x = k : _ x k^- (x^2 - ax + b)[x] = (k^2 - ak + b)(k - 1) RHL at x = k : _ x k^+ (x^2 - ax + b)[x] = (k^2 - ak + b)k Equating LHL and RHL: (k^2 - ak + b)(k - 1) = (k^2 - ak + b)k k^2 - ak + b = 0 Since f(x) is continuous at exactly two integers x = 2 and x = 3 , these must be the roots of the quadratic equation x^2 - ax + b = 0 . Using the properties of roo

Let [x] denote the greatest integer less than or equal to x . If the function f(x) = (x^2 - ax + b)[x] is continuous at exactly two integers x = 2 and x = 3 , where a and b are real constants, then the value of a^2 + b^2 is _____.

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