Functions 1 Question 13
14. The values of $f(x)=3 \sin \sqrt{\frac{\pi^{2}}{16}-x^{2}}$ lie in the interval…
(1983, 2M)
True/False
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Answer:
Correct Answer: 14. $0, \frac{3}{\sqrt{2}}$
Solution:
- Given, $f(x)=3 \sin \sqrt{\frac{\pi^{2}}{16}-x^{2}}$
$\Rightarrow$ Domain $\in-\frac{\pi}{4}, \frac{\pi}{4}$
$\therefore$ For range, $f^{\prime}(x)=3 \cos \sqrt{\frac{\pi^{2}}{16}-x^{2}} \cdot \frac{1(-2 x)}{2 \sqrt{\frac{\pi^{2}}{16}-x^{2}}}=0$
Where, $\quad \cos \sqrt{\frac{\pi^{2}}{16}-x^{2}}=0 \quad$ or $\quad x=0$
neglecting $\cos \sqrt{\frac{\pi^{2}}{16}-x^{2}}=0 \Rightarrow \frac{\pi^{2}}{16}-x^{2}=\frac{\pi^{2}}{4}$
$\Rightarrow x^{2}=-\frac{3 \pi^{2}}{16}$, never possible
$\Rightarrow \quad x=0$
Thus,
$$ f(0)=3 \sin \frac{\pi}{4}=\frac{3}{\sqrt{2}} $$
and
$$ f-\frac{\pi}{4}=f \frac{\pi}{4}=0 $$
Hence, $\quad$ range $\in 0, \frac{3}{\sqrt{2}}$
