Math, asked by kawal3169, 1 year ago

1. The maximum value of f(x) = sin4 x + cos4 x, is ________.

Answers

Answered by abhi569
1

Answer:

1.

Step-by-step explanation:

⇒ sin^4 x + cos^4 x

⇒ sin^4 x + ( cos^2 x )^2

⇒ sin^4 x + ( 1 - sin^2 x )^2           { cos^2 A = 1 - sin^2 A }

⇒ sin^4 x + 1 + sin^4 x - 2sin^2 x

⇒ 2sin^4 x - 2 sin^2 x + 1

⇒ 2sin^2 x( sin^2 x - 1 ) + 1

⇒ 2sin^2 x ( - cos^2x ) + 1

⇒ - 2sin^2 x . cos^2 x + 1

⇒ - 2( sin^2 x . cos^2 x ) + 1

⇒ - ( 4sin^2 x . cos^2 x ) / 2 + 1

⇒ - ( 2sinx cosx )^2 / 2 + 1             { 2 sinAcosA = sin2A }

⇒ - ( sin2x )^2 / 2 + 1

 For any angle, - 1 ≤ sinA ≤ 1

                           0 ≤ sin^2 x ≤ 1

⇒ - 0 / 2 + 1       or        - 1 / 2 + 1

⇒ 1                 or 1 / 2

Hence the maximum possible value of sin^4 x + cos^4 x is 1.

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