The number of points in (-infty, infty) for w | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $f(x) = x^2 - x \sin x - \cos x$.We want to find the number of roots of $f(x) = 0$.First,calculate the derivative: $f'(x) = 2x - (\sin x + x \

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) Let $f(x) = x^2 - x \sin x - \cos x$.We want to find the number of roots of $f(x) = 0$.First,calculate the derivative: $f'(x) = 2x - (\sin x + x \cos x) + \sin x = 2x - x \cos x = x(2 - \cos x)$.Since $-1 \leq \cos x \leq 1$,we have $2 - \cos x \geq 1 > 0$ for all $x$.Thus,$f'(x) > 0$ for $x > 0$,$f'(x) This means $f(x)$ is strictly decreasing on $(-\infty, 0]$ and strictly increasing on $[0, \infty)$.The global minimum occurs at $x = 0$,where $f(0) = 0^2 - 0 \sin(0) - \cos(0) = -1$.Since the minimum value is $-1$ and the function is continuous,we check the limits as $x 	o \pm \infty$.As $x 	o \pm \infty$,$f(x) \approx x^2 	o \infty$.Since $f(0) = -1 Similarly,since $f(0) = -1 Therefore,there are exactly $2$ points where $f(x) = 0$.

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

Similar Questions

The lanthanide contraction is responsible for the fact that

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta(t) = 2t^3 - 6t^2$. The torque on the wheel becomes zero at $t = $ ....... $\text{sec}$.

Half-life of a radioactive substance is $20 \, min$. The difference between the points of time when it is $33 \%$ disintegrated and $67 \%$ disintegrated is approximately .......... $min$.

In the measurement of a physical quantity $X = \frac{A^2 B}{C^{1/3} D^3}$,the percentage errors in the measurements of $A, B, C$,and $D$ are $2\%, 2\%, 4\%$,and $5\%$,respectively. The minimum percentage error in the measurement of $X$ is contributed by:

Vedclass Test Series

Exam Paper Generator

Online Exam Module