Let \( f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R \) be a continuous function su... VIDEO
Let \( f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R \) be a continuous function such that \( f(0) \) \( =1 \) and \( \int_{0}^{\pi / 3} f(t) d t=0 \)
Then which of the following statements is (are) TRUE?
(a) The equation \( f(x)-3 \cos 3 x=0 \) has at least one solution in \( \left(0, \frac{\pi}{3}\right) \)
(b) The equation \( f(x)-3 \sin 3 x=-\frac{6}{\pi} \) has at least one solution in \( \left(0, \frac{\pi}{3}\right) \)
(c) \( \lim _{x \rightarrow \infty} \frac{x \int_{0}^{x} f(t) d t}{1-e^{x^{2}}}=-1 \)
(d) \( \lim _{x \rightarrow \infty} \frac{\sin x \int_{0}^{x} f(t) d t}{x^{2}}=-1 \)
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