NIMCET 2016 — Mathematics PYQ

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Question

NIMCET 2016 — Mathematics PYQ

NIMCET | Mathematics | 2016

The number of points in $(- \infty, \infty)$ for which $x^{2} - x sin x - cos x = 0$ is

Choose the correct answer:

Explanation

Concept:
$\frac{d ( sin x )}{d x} = cos x$
$\frac{d ( cos x )}{d x} = - sin x$

$\frac{d ( uv )}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$

Calculation:

Let $f (x) = x^{2} - x sin x - cos x$

Differentiate with respect to $x$ , we get
$f^{'} (x) = 2 x - x cos x - sin x + sin x = x (2 - cos x)$

Differentiate the equation and compare to zero
$x (2 - cos x) = 0$

Maximum value of $cos x = 1$
So, $(2 - \cos x) > 0$

$f^{'} (x)$ is increasing when $x > 0$ ;
$f^{'} (x)$ is decreasing when $x < 0$

Therefore, $f (\infty) = \infty$ , $f (- \infty) = - \infty$ and $f (0) = - 1$

It will cut x-axis at 2 points. Hence 2 solutions.