Q140 / 200 · WBJEE 2013 · Mathematics
Which of the following real valued functions is/are not even functions?
- A$f(x)=x^{3} \sin x$
- B$f(x)=x^{2} \cos x$
- Correct answerC$f(x)=e^{x} x^{3} \sin x$
- Correct answerD$f(x)=x-[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$
Solution
We know that, if $f(-x)=f(x)$, then function is even and $f(-x)=-f(x)$, then function is odd.
(a) $f(x)=x^{3} \sin x$
$$
\begin{aligned}
f(-x) &=(-x)^{3} \sin (-x) \\
&=-x^{3}(-\sin x) \\
&=x^{3} \sin x=f(x)
\end{aligned}
$$
So, $f(x)$ is even.
(b) $f(x)=x^{2} \cos x$
$$
f(-x)=(-x)^{2} \cos (-x)=x^{2} \cos x=f(x)
$$
So, $f(x)$ is even.
(c) $f(x)=e^{x} x^{3} \sin x$
$$
\begin{array}{r}
f(-x)=e^{-x}(-x)^{3} \sin (-x) \\
=e^{-x} x^{3} \sin x \neq f(x)
\end{array}
$$
$\therefore \quad f(x)$ is not even.
(d) $f(x)=x-[x]$ $f(-x)=(-x)-[-x] \neq f(x)$
$\therefore f(x)$ is not even.
Hence, the correct answer is options: 'C', 'D'.