On which of the following intervals is the function f(x)=2x2−log|x|,x≠0 increasing in
A
(12,∞)
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B
(−∞,−12)∪(12,∞)
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C
(−∞,−12)∪(0,12)
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D
(−12,0)∪(12,∞)
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Solution
The correct option is D(−12,0)∪(12,∞) f(x)=2x2−log|x| ={2x2−logxx>02x2−log(−x)x<0 ⇒f′(x)=4x−1x,∀x≠0 For f(x) to be increasing f′(x)>0⇒4x−1x>0 ⇒4x2−1x>0 ⇒(2x−1)(2x+1)x>0 ⇒x(2x−1)(2x+1)>0 ⇒xϵ(−12,0)∪(12,∞).