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Let f : R → R be a function defined by f(x) = (x - 3)^{n1} (x - 5)^{n2}, n1,n2 ∈ N. Then, which of the following is NOT true? (1) For n1 = 3, n2 = 4, there exists α ∈ (3, 5) where f attains local maxima. (2) For n1 = 4, n2 = 3, there exists α ∈ (3, 5) where f attains local minima. (3) For n1 = 3, n2 = 5, there exists α ∈ (3, 5) where f attains local maxima. (4) For n1 = 4, n2 = 6, there exists α ∈ (3, 5) where f attains local maxima.
- (1) For n1 = 3, n2 = 4, there exists α ∈ (3, 5) where f attains local maxima.
- (2) For n1 = 4, n2 = 3, there exists α ∈ (3, 5) where f attains local minima.
- (3) For n1 = 3, n2 = 5, there exists α ∈ (3, 5) where f attains local maxima.
- (4) For n1 = 4, n2 = 6, there exists α ∈ (3, 5) where f attains local maxima.
Correct answer: (3) For n1 = 3, n2 = 5, there exists α ∈ (3, 5) where f attains local maxima.
Solution
The statement is incorrect because with n1 = 3 and n2 = 5, the function has a higher multiplicity at x = 5, which means it will not attain a local maximum in the interval (3, 5) as it approaches zero without changing sign.
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