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