Evaluate `int_(0)^(pi/2) cosx` `dx`

Answer

`(A)to(r);(B)to(s);(C)to(t);(D)to(p,q,r,s,t)`

Explanation

(A)`:.AMgeGM`
`:.(ab^(2)+ac^(2)+bc^(2)+ba^(2)+ca^(2)+cb^(2))/(6)ge(ab^(2)*ac^(2)*bc^(2)*ba^(2)*ca^(2)*cb^(2))^((1)/(6))=abc`
`:.a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2))ge6abc`
`implies lambda=6`
(B) `:.AMgeGM`
For b,c, we get `((b+c))/(2)gesqrt(bc)`
`implies (b+c)ge2sqrt(bc)" " "....."`
For c,a, we get
`((c+a))/(2)ge sqrt(ca)`
`implies (c+a)ge 2sqrt(ca)" " ".....(ii)"`
and a, b, we get
`((a+b))/(2)ge sqrt(ab)`
`implies (a+b)ge 2sqrt(ab)" " ".....(iii)"`
On multiplying Eqs. (i), (ii) and (iii), we get
`(b+c)(c+a)(a+b)ge8abc`
`implies (1-a)(1-b)(1-c)ge 8abc [:. a+b+c=1]`
`implies ((1)/(a)-1)((1)/(b)-1)((1)/(c)-1)ge 8`
`:. lambda=8`
(C ) `:.AMgeHM`
`((s-a)+(s-b)+(s-c))/(3)ge(3)/((1)/(s-a)+(1)/(s-b)+(1)/(s-c))`
`implies (3s-(a+b+c))/(3)ge(3)/((1)/(s-a)+(1)/(s-b)+(1)/(s-c))`
`implies (3s-s)/(3)ge(3)/(((1)/(s-a)+(1)/(s-b)+(1)/(s-c)))`
`implies(2s)/(s-a)+(2s)/(s-b)+(2s)/(s-c)ge9`
Here, `lambda-1=9`
`:. lambda =10`
(D) If a, b, c are in GP.
`a^(lambda),b^(lambda),c^(lambda)` are also in GP.
Then, `AMge GM`
`(a^(lambda)+c^(lambda))/(2)geb^(lambda)`
`impliesa^(lambda)+c^(lambda)ge2b^(lambda)`
`implies((a)/(b))^(lambda)+((c)/(b))^(lambda)ge2`
`:. lambdainR`
Hence, `lambda=2,4,6,8,10`