Number of Real Roots of Trigonometric Transcendental Equation

To find the total number of real roots of the equation $f(x) = x^2 - x \sin x - \cos x = 0$ in the interval $(-\infty, \infty)$, we analyze the behavior and properties of the function $f(x)$.

Step 1: Symmetry of the Function

First, we check the symmetry of $f(x)$: $f(-x) = (-x)^2 - (-x)\sin(-x) - \cos(-x) = x^2 - (-x)(-\sin x) - \cos x = x^2 - x\sin x - \cos x = f(x)$ Thus, $f(x)$ is an even function, which means its graph is symmetric with respect to the $y$-axis. Any positive root will have a corresponding negative root of the same magnitude.

Step 2: Critical Points and Monotonicity

Differentiating $f(x)$ with respect to $x$: $f'(x) = \frac{d}{dx}\left(x^2 - x \sin x - \cos x\right)$ $f'(x) = 2x - (\sin x + x \cos x) - (-\sin x)$ $f'(x) = 2x - \sin x - x \cos x + \sin x$ $f'(x) = 2x - x \cos x = x(2 - \cos x)$

Since $-1 \le \cos x \le 1$ for all $x \in \mathbb{R}$, we have: $1 \le 2 - \cos x \le 3$ Thus, $(2 - \cos x) > 0$ for all $x \in \mathbb{R}$.

This implies that the sign of $f'(x)$ is completely determined by $x$:

• For $x > 0$, $f'(x) > 0 \implies f(x)$ is strictly increasing on $[0, \infty)$.
• For $x < 0$, $f'(x) < 0 \implies f(x)$ is strictly decreasing on $(-\infty, 0]$.
• At $x = 0$, $f'(0) = 0$, representing a global minimum.

Step 3: Determining the Number of Roots

Evaluating $f(x)$ at $x = 0$: $f(0) = 0^2 - 0 \cdot \sin(0) - \cos(0) = -1 < 0$

Taking the limit as $x \to \infty$: $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x^2 - x\sin x - \cos x) = \infty \quad (\text{since } f(x) \ge x^2 - x - 1)$

• Since $f(0) = -1 < 0$, $\lim_{x \to \infty} f(x) = \infty$, and $f(x)$ is strictly increasing and continuous on $[0, \infty)$, by the Intermediate Value Theorem, there exists exactly one root in the interval $(0, \infty)$.
• Since $f(x)$ is an even function, there exists exactly one root in the interval $(-\infty, 0)$.
• At $x = 0$, $f(0) = -1 \neq 0$, so $x = 0$ is not a root.

Therefore, the total number of points satisfying the equation is: $1 + 1 = 2$

Correct Answer: (C) 2