Application of Derivatives — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives2 Marks
Question
At what points in the interval [0, 2$\pi$], does the function sin 2x attain its maximum value?
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Answer
Given that f(x) = sin 2x, $x \in[0,2 \pi]$ $f^\prime$(x) = 2 cos 2x Now, f'(x) = 0 $\Rightarrow$ cos 2x = 0 $\Rightarrow$ 2x = 0 $\Rightarrow x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ Now, we evaluate the value of f at critical point $x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ and at end points of the interval [0, 2$\pi$] $f\left(\frac{\pi}{4}\right)=\sin \frac{\pi}{2}=1$ $f\left(\frac{3 \pi}{4}\right)=\sin \frac{3 \pi}{2}=-1$ $f\left(\frac{5 \pi}{4}\right)=\sin \frac{5 \pi}{2}=1$ $f\left(\frac{7 \pi}{4}\right)=\sin \frac{7 \pi}{2}=-1$ f(0) = sin 0, and f(2$\pi$) = sin 4$\pi$ = 0 Therefore, the absolute maximum value of f on $[0,2 \pi]$ is 1 occuring at $x=\frac{\pi}{4}$ and $x=\frac{5 \pi}{4}$.
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