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For the function f(x) = 2x² - log|x|, with x ≠ 0, on which interval is the function increasing throughout?
- (1/2, ∞)
- (-∞, -1/2) ∪ (1/2, ∞)
- (-∞, -1/2) ∪ (0, 1/2)
- (-1/2, 0) ∪ (1/2, ∞)
Correct answer: (-1/2, 0) ∪ (1/2, ∞)
Solution
f'(x)=4x-1/x=(4x^2-1)/x. For x>0 this is positive when x>1/2; for x<0 the numerator 4x^2-1<0 and denominator<0 make it positive when -1/2<x<0. So f increases on (-1/2,0) U (1/2, infinity).
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