Let f(x) = x^2 + x sin x - cos x. Then | vedclass

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(A) Given $f(x) = x^2 + x \sin x - \cos x$.We find the derivative: $f'(x) = 2x + (\sin x + x \cos x) + \sin x = 2x + 2 \sin x + x \cos x = x(2 + \cos

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$f(x) = 0$ has at least one real root

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(A) Given $f(x) = x^2 + x \sin x - \cos x$.We find the derivative: $f'(x) = 2x + (\sin x + x \cos x) + \sin x = 2x + 2 \sin x + x \cos x = x(2 + \cos x) + 2 \sin x$.For $x > 0$, $f'(x) = x(2 + \cos x) + 2 \sin x$. Since $2 + \cos x > 0$ and $x > 0$, and $2 \sin x$ is not always negative, we analyze the behavior. Alternatively, note that $f(0) = -1$.As $x 	o \infty$, $f(x) 	o \infty$. Since $f(0) = -1 Similarly, as $x 	o -\infty$, $f(x) 	o \infty$. Since $f(0) = -1 Thus, $f(x) = 0$ has at least two real roots (one positive and one negative), which implies it has at least one real root.

Let $f(x) = x^2 + x \sin x - \cos x$. Then

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