x2cosx

x^2 cos x, written as f(x) = x^2 cos x, is a real-valued function defined for all real numbers x. It is the product of the square of x and the cosine of x, and the function is entire, since both x^2 and cos x are analytic everywhere.

Because cos x is even and x^2 is even, f is an even function, satisfying f(-x) = f(x). The zeros occur at x = 0 and at x = π/2 + kπ for integers k, since cos x = 0 there. At x = 0 the factor x^2 also ensures a zero of multiplicity two.

Derivatives of f are straightforward from the product rule. The first derivative is f'(x) = 2x cos x − x^2 sin x. The second derivative is f''(x) = 2 cos x − 4x sin x − x^2 cos x, which can be used to study curvature and critical points.

The Maclaurin series of f is obtained by multiplying the series for cos x by x^2: f(x) = x^2 cos x = Σ_{n=0}^∞ (−1)^n x^{2n+2} / (2n)!. The first terms are x^2 − x^4/2! + x^6/4! − x^8/6! + …

Behaviorally, the amplitude of f grows like x^2 while it oscillates with the cosine factor, producing infinitely many zeros and an unbounded range. The function is used in contexts involving polynomial growth modulated by oscillatory components and appears in various integrals and analyses in mathematics.