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

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)