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Math Help - Find maximum and minimum value of

  1. #1
    Senior Member pankaj's Avatar
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    Find maximum and minimum value of

    Find maximum and minimum value of

     <br />
f(x)=|1+2\cos x|+|1+2\sin x|,<br />
x\in R<br />
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  2. #2
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    Quote Originally Posted by pankaj View Post
    Find maximum and minimum value of

     <br />
f(x)=|1+2\cos x|+|1+2\sin x|,<br />
x\in R<br />
    you only need to consider the interval [0, 2\pi]. now \cos x = \frac{-1}{2} at x = \frac{2 \pi}{3},\ \frac{4 \pi}{3} and \sin x = \frac{-1}{2} at x=\frac{7 \pi}{6}, \ \frac{11 \pi}{6}. thus:

    |1 + 2\cos x|=1 + 2 \cos x, \ |1 + 2\sin x|=1+2 \sin x for any x \in [0, 2 \pi/3] \cup [11 \pi/6, 2\pi],

    |1 + 2\cos x|=-1 - 2 \cos x, \ |1 + 2\sin x|=1+2 \sin x for any x \in [2 \pi/3, 7\pi/6],

    |1 + 2\cos x|=-1 - 2 \cos x, \ |1 + 2\sin x|=-1-2 \sin x for any x \in [7 \pi/6, 4 \pi/3], and finally:

    |1 + 2\cos x|=1 + 2 \cos x, \ |1 + 2\sin x|=-1-2 \sin x for any x \in [4 \pi/3, 11 \pi/6].

    now use the indentity \sin x \pm \cos x = \sqrt{2} \sin(x \pm \pi/4) and the rest is easy. the maximum is quickly seen to be 2(1+\sqrt{2}).
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