3IfA- cos'0 + sin'o, than prove that for all values of0, 4 SAS

1.	3IfA- cos'0 + sin'o, than prove that for all values of0, 4 SAS
Answer» A= sin⁴∅+cos²∅ = sin⁴∅ +1-sin²∅ for maximum and minimum value f'(∅) =0 => 4sin³∅cos∅-2sin∅cos∅=0=> 2sin∅cos∅(2sin²∅-1) = 0=> sin2∅(-cos2∅) = 0=> 1/2*(sin4∅) = 0 => 4∅ = 0 ∅ = 0, π/4, π/2 ... now value of A at ∅ = 0 is sin0+cos0 = 1and value at ∅ = π/4 is (1/√2)⁴+(1/√2)² = 1/4+1/2= 3/4value at ∅ = π/2 = 1+0 = 1 so, minimum Value is = 3/4 and maximum Value is 1

Answer»

A= sin⁴∅+cos²∅

= sin⁴∅ +1-sin²∅

for maximum and minimum value f'(∅) =0

=> 4sin³∅cos∅-2sin∅cos∅=0=> 2sin∅cos∅(2sin²∅-1) = 0=> sin2∅(-cos2∅) = 0=> 1/2*(sin4∅) = 0

=> 4∅ = 0 ∅ = 0, π/4, π/2 ...

now value of A at ∅ = 0 is sin0+cos0 = 1and value at ∅ = π/4 is (1/√2)⁴+(1/√2)² = 1/4+1/2= 3/4value at ∅ = π/2 = 1+0 = 1

so, minimum Value is = 3/4 and maximum Value is 1

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