MCQ
Consider the function $f (x) = x\, \cos x - \sin x$, then identify the statement which is correct .
- A$f$ is neither odd nor even
- B$f$ is monotonic decreasing at $x = 0$
- C$f$ has a maxima at $x = \pi$
- D$f$ has a minima at $x = - \pi$
✓
Answer
$f ' (x) = - x \sin x = 0$ when $x = 0$ or $\pi$
$\left. \begin{array}{l}f ' ({0^ - }) = ( - )( - )( - ) < 0\\f'\,({0^ + }) = ( - )( + )( + ) < 0\end{array} \right]$ no sign change
This also implies that $f$ is decreasing at $x = 0\ (B)$ is correct
$f''(x) = - (x \cos x + \sin x)$
$f'' (\pi ) = - (--\pi ) > 0$ minima at $x = \pi $
$f '' (- \pi ) = - (\pi ) < 0$ maxima at $x = \pi $]
$\left. \begin{array}{l}f ' ({0^ - }) = ( - )( - )( - ) < 0\\f'\,({0^ + }) = ( - )( + )( + ) < 0\end{array} \right]$ no sign change
This also implies that $f$ is decreasing at $x = 0\ (B)$ is correct
$f''(x) = - (x \cos x + \sin x)$
$f'' (\pi ) = - (--\pi ) > 0$ minima at $x = \pi $
$f '' (- \pi ) = - (\pi ) < 0$ maxima at $x = \pi $]
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