f(x)=x2cosx

Assuming you mean f(x) = x^2 cos x, here are handy facts and formulas for exam prep.

• Domain and basic properties

  • Domain: all real numbers.
  • f is even: f(-x) = f(x).

• First derivative (useful for increasing/decreasing and critical points)

  • f'(x) = 2x cos x − x^2 sin x
  • Critical points satisfy f'(x) = 0:
    • x = 0, or
    • 2x cos x − x^2 sin x = 0 ⇒ x(2 cos x − x sin x) = 0 ⇒ tan x = 2/x (for x ≠ 0)
  • There are infinitely many critical points, solutions of tan x = 2/x.

• Second derivative (concavity and inflection)

  • f''(x) = derivative of (2x cos x − x^2 sin x)
  • f''(x) = (2 − x^2) cos x − 4x sin x
  • Inflection points satisfy f''(x) = 0 (no closed-form solution; numerical).

• Maclaurin series (useful for small-x behavior)

  • cos x = 1 − x^2/2 + x^4/24 − …
  • f(x) = x^2 cos x = x^2 − (1/2)x^4 + (1/24)x^6 − …
  • In particular, near 0, f(x) ~ x^2 (positive for small x ≠ 0), with f(0) = 0.

• Local behavior at 0

  • f(0) = 0, f''(0) = 2 > 0, so x = 0 is a local minimum (and the function is nonnegative near 0).

• Antiderivative (indefinite integral)

  • ∫ x^2 cos x dx = x^2 sin x + 2x cos x − 2 sin x + C

• Large-scale behavior

  • As |x| → ∞, f(x) = x^2 cos x oscillates with amplitude growing like x^2 (no horizontal asymptote).

If you want, tell me the exact task you’re studying (e.g., find increasing/decreasing intervals, locate inflection points numerically, compute a Taylor polynomial to a certain degree, or solve a specific problem), and I’ll tailor the steps and, if needed, provide approximate roots or plots.