Q52 / 91 · KVPY 2020 (SB/SX) · Mathematics

Let $f (x) = \sin x + (x^{3} - 3 x^{2} + 4 x - 2) \cos x$ for $x \in (0, 1) .$ Consider the following statements
I. $f$ has a zero in $(0, 1)$
II. $f$ is monotone in $(0, 1)$
Then

Solution

$f (x) = \sin x + (x^{3} - 3 x^{2} + 4 x - 2) \cos x, x \in (0, 1)$

$x \in (0, 1)$

$f (0) = - 2 > 0$

$f (1) = \sin 1 < 0$

$∵ f (0) \cdot f (1) < 0 \Rightarrow f (x)$ has a zero in $(0, 1)$

Now,

$f (x) = \sin x + [(x - 1)^{3} + (x - 1)] \cos x$

$\Rightarrow f^{'} (x) = (3 (x - 1)^{2} + 2) \cos x - \sin x [(x - 1)^{3} + (x - 1)]$

$= [3 (x - 1)^{2} + 2] \cos x + [(1 - x)^{3} + (1 - x)] \sin x$

$> 0 \forall x \in (0, 1)$

$\Rightarrow f (x)$ is monotone in $(0, 1)$

Hence, the correct answer is option: 'A'.