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If f(x)={{:(-e^(-x)+k,",",xle0),(e^(x)+1...

If `f(x)={{:(-e^(-x)+k,",",xle0),(e^(x)+1,",",0ltxlt1),(ex^(2)+lambda,",",xge1):}` is one-one and monotonically increasing `AA x in R`, then

A

maximum value of k is 1

B

maximum value of k is 3

C

minimum value of `lambda` is 0

D

minimum value of `lambda` is 1

Text Solution

Verified by Experts

The correct Answer is:
B, D
If `f(x)={{:(-e^(-x)+k",",xle0),(e^(x)+1",",0ltxlt1),(ex^(2)+lambda",",xge1):}`
`y=-e^(-x)+k` is increasing for `x lt0`
`y=e^(x)+1` is increasing for `0ltxlt1`
`y=ex^(2)+lambda` is increasing for `xgt1`
`therefore" f(x) is increasing for " AA x in R`
`rArr" "f(0^(-))lef(0^(+))`
`rArr" "-1+k le 1+1`
`rArr" "kle3`
and `f(1^(-))lef(1^(+))`
`rArr" "e+1le e +lambda`
`rArr" "lambda le 1`
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