Which of the following real valued function is/are not even functions?
f(x)=x2cosx
f(x)=x3sinx
f(x)=exx3sinx
f(x)=x−[x], where [x] denotes the greatest integer less than or equal to x
A
f(x)=x−[x], where [x] denotes the greatest integer less than or equal to x
B
f(x)=x3sinx
C
f(x)=x2cosx
D
f(x)=exx3sinx
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Solution
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We know that, if f(−x)=f(x) then function is even and if f(−x)=−f(x) then function is odd (a) f(x)=x3sinx f(−x)=(−x)3sin−x −x3(−sinx) =x3sinx=f(x) So, f(x) is even
(b) f(x)=x2cosx f(−x)=(−x)2cos−x=x2cosx=f(x) So, f(x) is even
(c) f(x)=exx3sinx f(−x)=e−x(−x)3sin−x =e−xx3sinx≠f(x) ∴f(x) is not even
(d)f(x)=x−[x] f(−x)=(−x)−[−x]≠f(x) ∴f(x) is not even.
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