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4sin 2 x+9cosec 2 x find out the minimum value of this function

4sin2x+9cosec2x find out the minimum value of this function 
 

Grade:12th pass

3 Answers

ayush
29 Points
7 years ago
Let sin^2x=k>=0.Thus given expression by am-gm inequality will have lowest value when sin^2x=cosec^2 x. Thus min value is 13.
jfjyf
17 Points
3 years ago
Let write equation as:
      4(sin2x + cosec2x) + 5cosec2x
Applying AM-GM inequality in sin2x + cosec2x
     (sin2x + cosec2x)/2 >=  (sin2x.cosec2x)1/2
=> (sin2x + cosec2x)  >=  2
So, minimum of (sin2x + cosec2x) is 2
which implies sin2x = cosec2x = 1.
Minimum of 
      4(sin2x + cosec2x) + 5cosec2x   =   4*2 + 5*1 = 13
ANUBHAV SHARMA
13 Points
2 years ago
Y=4sin^2x+9cosec^2xSolutionBy making whole square4sin^2x+9cosec^2x+12-12(2sinx-3cosecx)^2+12(2sinx-3cosecx)^2>=0(2sinx-3cosecx)^2+12>=12Hence the min value of y is 12Y=[12,infinity)

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