Mathematics

Mathematics

Problem

Let f:[−π2,π2]→Rf: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to \mathbb{R} be a continuous function such that f(0)=1f(0) = 1 and ∫0π/3f(t)dt=0\int_{0}^{\pi/3} f(t) dt = 0 Then which of the following statements is(are) TRUE ? [JEE-Advanced 2021] (A) The equation f(x)−3cos⁡3x=0f(x) - 3 \cos 3x = 0 has at least one solution in (0,π3)\left(0, \frac{\pi}{3}\right) (B) The equation f(x)−3sin⁡3x=−6πf(x) - 3 \sin 3x = -\frac{6}{\pi} has at least one solution in (0,π3)\left(0, \frac{\pi}{3}\right) (C) lim⁡x→0∫0xf(t)dt1−ex2=−1\lim_{x \to 0} \frac{\int_{0}^{x} f(t) dt}{1 - e^{x^2}} = -1 (D) lim⁡x→0∫0sin⁡xf(t)dtx2=−1\lim_{x \to 0} \frac{\int_{0}^{\sin x} f(t) dt}{x^2} = -1

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