Question

find maximum and minimum value of following expression. 1. cos²x+2(sinx+cosx)² 2. 3sin²x-5(cosx-sinx)²+3​

Answer 1

Answer:

We know that AM≥GM

∴

2

2

sinx

+2

cosx

≥

2

sinx

2

cosx

⇒2

sinx

+2

cosx

≥2

2

sinx+cosx

⇒2

sinx

+2

cosx

≥2×2

2

sinx+cosx

⇒2

sinx

+2

cosx

≥2

1+

2

sinx+cosx

But sinx+cosx=

2

sin(x+

4

π

)≥−

2

∴2

sinx

+2

cosx

≥2

1−

2

2

⇒2

sinx

+2

cosx

≥2

1−

2

1

,∀x∈R

Hence, minimum value is 2

1−

2

1