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Find the minimum value of 3 sin power 6 x + 3 cos power 6 x

Find the minimum value of 3sin power 6 x + 3cos power 6 x

Grade:11

2 Answers

Deep Chand
42 Points
3 years ago
We have 3^{sin^6x}+3^{cos^6x}   Now
We know that 
A.M.\ge G.M
\frac{3^{sin^6x}+3^{sin^6x}}{2}\ge\sqrt{3^{sin^6x}.3^{cos^6x}}
\frac{3^{sin^6x}+3^{sin^6x}}{2}\ge\sqrt{3^{(sin^6x+cos^6x)}}
\frac{3^{sin^6x}+3^{sin^6x}}{2}\ge\sqrt{3^{(1-3sin^2x.cos^2x)}}
\frac{3^{sin^6x}+3^{sin^6x}}{2}\ge\sqrt{3^{(1-\frac{3}{4}sin^22x)}}
If {3^{sin^6x}+3^{sin^6x}} will be minimum only when sin^22x  should be maximum 0\le sin^22x \le1 , then sin^22x =1
Now 
{3^{sin^6x}+3^{sin^6x}}\ge 2 \sqrt {3^{1-\frac {3}{4}}}
{3^{sin^6x}+3^{sin^6x}}\ge 2 \sqrt {3^{\frac {1}{4}}}
Hence minimum value of {3^{sin^6x}+3^{sin^6x}} =(768)^\frac {1}{8}
 
DEEP CHAND SAROJ 
B.Tech (IIT -DELHI)
Deep Chand
42 Points
3 years ago
We have 3^{sin^6x}+3^{cos^6x}   then 
We know that 
A.M.\ge G.M.
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{sin^6x}.3^{cos^6x}}
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{sin^6x+cos^6x}}
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{1-3sin^2x.cos^2x}}
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{1-\frac{3}{4}sin^22x}}
For {3^{sin^6x}+3^{cos^6x}} minimum then sin^22x should be maximum
We know that 0\le sin^22x\le 1 then
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{1-\frac{3}{4}}}
\frac{3^{sin^6x}+3^{cos^6x}}{2} \ge \sqrt {3^{\frac{1}{4}}}
{3^{sin^6x}+3^{cos^6x}} \ge 2\sqrt {3^{\frac{1}{4}}}
{3^{sin^6x}+3^{cos^6x}} \ge (768)^\frac {1}{8}
Minimum value of {3^{sin^6x}+3^{cos^6x}} = (768)^\frac {1}{8}
DEEP CHAND SAROJ 
B.Tech (IIT-DELHI)
 

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