1.

Statement-1 e^(x)+e^(-x) gt 2 +x^2 is an increasing function on R.

Answer»

Statement-1 TRUE statement -1 is True,Statement -2is True statement -2 is a CORRECT explanation for Statement-3
Statement-1 True statement -1 is True,Statement -2is True statement -2 is not a correct explanation for Statement-3
Statement-1 True statement -1 is True,Statement -2is False
Statement-1 is False ,Statement -2 is True

Solution :We have
`f(x)=e^x+e^(-x)-2-x^2`
`rArr f(x)=e^x-e^(-x)-2X`
`rArr f(x)= e^x+e^(-x)-2=((e^x-1)^2)/(e^x)gt 0 `for all `x ne 0`ltbr gt `rArr ` f(x) in INCREASING in R
`rArr f(x) gtf(0) " for all " x in R , x ne 0 `
`f(x) gt 0 "for all " x (ne 0) in R`
`f (x) gt f(0) " for all " x ne 0 `
`e^x+e^(-x)-2-x^2 LT 0 " for all " x ne 0 `
`rArr e^x+e^(-x) gt 2+ x^2 " for all " x ne 0`
Hence both the statements are true statement-2 is a correct explanation of statment-1


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