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 $f(x) = x^2 - x \sin x - \cos x$.Then,the derivative is $f'(x) = 2x - (\sin x + x \cos x) - (-\sin x) = 2x - x \cos x = x(2 - \cos x)$.Since $

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) Let $f(x) = x^2 - x \sin x - \cos x$.Then,the derivative is $f'(x) = 2x - (\sin x + x \cos x) - (-\sin x) = 2x - x \cos x = x(2 - \cos x)$.Since $-1 \le \cos x \le 1$,we have $2 - \cos x > 0$ for all real $x$.Thus,$f'(x) = 0$ only at $x = 0$.For $x For $x > 0$,$f'(x) > 0$,so $f(x)$ is strictly increasing.Therefore,$x = 0$ is the point of global minimum.The minimum value is $f(0) = 0^2 - 0 \sin(0) - \cos(0) = -1$.As $x 	o \infty$,$f(x) 	o \infty$,and as $x 	o -\infty$,$f(x) 	o \infty$.Since $f(x)$ is continuous,it crosses the $x$-axis exactly once in $(-\infty, 0)$ and exactly once in $(0, \infty)$.Thus,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

If $f(x)=3x^3-9x^2-27x+15$,then the maximum value of $f(x)$ is $.....$

Let $f(x) = \begin{cases} 3 - x, & 0 \le x < 1 \\ x^2 + \log_e b, & x \ge 1 \end{cases}$. The set of values of $b$ such that $f(x)$ has a local minimum at $x = 1$ is

For the function $f(x) = \int_{0}^{x} \frac{\sin t}{t} dt$,where $x > 0$,which of the following is true?

An open tank with a square bottom is to contain $4000 \ cm^3$ of liquid. Find the dimensions of the tank such that the surface area of the tank is minimum.

Vedclass Test Series

Exam Paper Generator

Online Exam Module