If f(x)=cosx[x2π]+12∀x∈[−2π,2π] and [⋅] represent greatest integer. Then which of the following is true
f(x) is an odd function
f(x) is an even function
f′(x)=2sinx∀−2π≤x<0
f′(x)=−2sinx∀0<x<2π
A
f(x) is an even function
B
f′(x)=2sinx∀−2π≤x<0
C
f(x) is an odd function
D
f′(x)=−2sinx∀0<x<2π
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Solution
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Case 1: −2π≤x<0 f(x)=cosx−1+12=−2cosx case 2: 0≤x<2π f(x)=cosx0+12=2cosx Clearly f(−x)=−f(x)⇒f(x) is an odd function. Option 'C' and 'D' are obviously correct.
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