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Question Number 107018 by bemath last updated on 08/Aug/20

Given f(x)=((10)/(2−sin 2x)). Find maximum value f(x).

Givenf(x)=102−sin2x.Findmaximumvaluef(x).

Commented by PRITHWISH SEN 2 last updated on 08/Aug/20

for f(x) to be max. 2−sin 2x must be min. i.e sin 2x must be max.i.e 1 ∴ max. f(x)=((10)/(2−1)) = 10 and for min. sin 2x must be min. i.e −1 ∴ min.f(x)=((10)/(2−(−1)))= 3.33..

forf(x)tobemax.2−sin2xmustbemin.i.esin2xmustbemax.i.e1∴max.f(x)=102−1=10andformin.sin2xmustbemin.i.e−1∴min.f(x)=102−(−1)=3.33..

Commented by kaivan.ahmadi last updated on 08/Aug/20

f is maximum if 2−sin2x is minimum.so since −1≤sin2x≤1⇒−1≤−sin2x≤1 ⇒1≤2−sin2x≤3⇒ min(2−sin2x)=1⇒ max(f)=((10)/1)=10

fismaximumif2−sin2xisminimum.sosince−1⩽sin2x⩽1⇒−1⩽−sin2x⩽1⇒1⩽2−sin2x⩽3⇒min(2−sin2x)=1⇒max(f)=101=10

Answered by bemath last updated on 08/Aug/20

we claim −1≤−sin 2x≤1 −1+2≤2−sin 2x≤2+1 1 ≤2−sin 2x≤3 ⇒(1/3)≤(1/(2−sin 2x))≤1 ((10)/3)≤((10)/(2−sin 2x))≤10 ⇒((10)/3)≤f(x)≤10 → { ((max=10)),((min=((10)/3))) :}

weclaim−1⩽−sin2x⩽1−1+2⩽2−sin2x⩽2+11⩽2−sin2x⩽3⇒13⩽12−sin2x⩽1103⩽102−sin2x⩽10⇒103⩽f(x)⩽10→{max=10min=103

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