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If m is the minimum value of k for which the function f(x) = x√(kx − x²) is increasing in the interval [0,3] and M is the maximum value of f in [0,3] when k = m, then the ordered pair (m, M) is equal to:
- (5, 3√6)
- (4, 3√3)
- (4, 3√2)
- (3, 3√3)
Correct answer: (4, 3√3)
Solution
The function f(x) is increasing when its derivative is non-negative. By analyzing the derivative and finding the minimum value of k that satisfies this condition in the interval [0,3], we determine that m is 4. Evaluating f at this k gives the maximum value M as 3√3.
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