MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

⇐ BackProceed to Checkout ⇒
There are no items in this cart.
Continue Shopping
Menu
Grade: 11 

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

Answers : (2)

Deep Chand
42 Points 
							
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)
3 years ago
Deep Chand
42 Points 
							
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)
 
3 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies

Other Related Questions on Trigonometry

please help me in prove that costheta+cosphi/sintheta+sinphi=sintheta-sinphi/cosphi-costheta Answer & Earn Cool Goodies
In a ABC, the point D divides BC in the ratio 1:2. Also AD is perpendicular to AB. If the value of... Answer & Earn Cool Goodies
If a=π/15 than proved that cos2acos4acos8acos14a=1/16
Which of the following is a homogeneous mixture? A) Milk B) Steel C) Smoke D) Soil
prove the following: cotA×cotB+cotB×cotC+cotC×cotA=1
View all Questions »


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 31 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers To Win!!! Click Here for details