consider f(X)= `cos2x+2xlambda^(2)+(2lambda+1)(lambda-1)x^(2),lambda in R`
If `alpha ne beta and f(alpha+beta)/(2)lt(f(alpha)+f(beta))/(2)` for `alpha` and `beta` then find the values of `lambda`

Answer

`(-oo,-1]cup[3//2,oo)`

Explanation

f(x)=`cos2x+2xlambda^(2)+(2lambda+1)(lambda-1)x^(2),lambda in R`
`therefore` of y =f(X) must be concave upward `forall x in R`
`f''(x) ge 0 forall x in R`
`rarr (2lambda+1)(lambda-1)ge2cos2xforall x in R`
`rarr 2 lambda^(2)-lambda-3ge0`
` rarr (2lambda-3)(lambda+1)ge0`
`rarr lambda in (-oo.-1]cup[(3)/(2),oo]`