Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to \mathbb{R}$ be a continuous function such that $f(0) = 1$ and $\int_{0}^{\frac{\pi}{3}} f(t)dt = 0$. Then which of the following statements is (are) **TRUE**?

$f \text{ continuous on } \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right], \qquad f(0) = 1$ $\int_{0}^{\pi/3} f(t)\,dt = 0$ $\text{Existence of roots in } (0,\pi/3); \text{ and two limits}$ $F(x) = \int_{0}^{x} f(t)\,dt; \quad F(0)=0,\ F\!\left(\tfrac{\pi}{3}\right)=0,\ F'(x)=f(x)$ $F'(0) = f(0) = 1$ $G(x) = F(x) - \sin 3x; \quad G(0) = 0, \ G\!\left(\tfrac{\pi}{3}\right) = 0$ $\Rightarrow \exists\, c \in \left(0,\tfrac{\pi}{3}\right):\ f(c) - 3\cos 3c = 0 \Rightarrow (1) \text{ TRUE}$ $G(x) = F(x) + \cos 3x + \tfrac{6}{\pi}x; \quad G(0) = 1, \ G\!\left(\tfrac{\pi}{3}\right) = 0 - 1 + 2 = 1$ $\Rightarrow \exists\, c:\ f(c) - 3\sin 3c = -\tfrac{6}{\pi} \Rightarrow (2) \text{ TRUE}$ $\text{domain bounded} \Rightarrow \text{the limits are } x \to 0$ $F(x) = x + o(x) \text{ as } x \to 0$ $\lim_{x \to 0} \dfrac{x\,F(x)}{1 - e^{x^{2}}} \sim \dfrac{x\cdot x}{-x^{2}} = -1 \Rightarrow (3) \text{ TRUE}$ $\lim_{x \to 0} \dfrac{\sin x\,F(x)}{x^{2}} \sim \dfrac{x\cdot x}{x^{2}} = 1 \neq -1 \Rightarrow (4) \text{ FALSE}$ $\boxed{\text{TRUE statements: } (1), (2), (3)}$ $F = \int_0^x f;\ \text{Rolle} \Rightarrow (1),(2);\ x \to 0 \Rightarrow (3) \text{ TRUE}, (4) \text{ FALSE}$