Let f:[-\frac{\pi}{2}, \frac{\pi}{2}] \to \mathbb{R} be a continuous function such that f(0)=1 \text{ and } \int_{0}^{\frac{\pi}{3}} f(t) d t=0 Then which of the following statements is (are) TRUE? [JEE Adv 2021 P2]

The equation f(x)-3 \cos 3 x=0 has at least one solution in (0, \frac{\pi}{3})

The equation f(x)-3 \sin 3 x=-\frac{6}{\pi} has at least one solution in (0, \frac{\pi}{3})

\lim_{x \to \infty} {x \int_{0}^{x} f(t) d t}/{1-e^{x 2}}=-1 `

\lim_{x \to \infty} {\sin x \int_{0}^{x} f(t) d t}/{x^{2}}=-1

Definite Integration

Section: Mathematics

Question : 11 of 46

Marks: +1, -0

Let f:[−‌

,‌

]→ℝ be a continuous function such that f(0)=1‌ and ‌

∫

‌

f(t)dt=0 Then which of the following statements is (are) TRUE?

[JEE Adv 2021 P2]

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