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Let f be a function from [a,b] to R , (w...

Let f be a function from `[a,b] to R` , (where `a, b in R` ) f is continuous and differentiable in [a, b] also f(a) = 5, and `f'(x) le 0` for all `x in [a,b]` then for all such functions f,`f(b) + f(lambda)` lies in the interval where `lambda in (a,b)`

A. `(oo, - 10]`
B. `(- oo, 20]`
C. `(-oo, 10]`
D. `(- oo, 5]`

A

`(-oo, 10]`

B

`(-oo,-20]`

C

`(-oo,10]`

D

`(-oo,5]`

Text Solution

Verified by Experts

The correct Answer is:
C
Applying LMVT on f in [a,b]
`(f(b) -f(a))/(b-a) = f(c) in [a,b]`
ATQ `(f(b) - f(a))/(b-a) le 0`
ie `" " f(b) - f(a) le0" ".....(i)`
agin using LMVT in `[a, lambda]`
`(f(lambda) - f(a))/(lambda -a) le 0`
`rArr f(x) - f(a) le 0 " " ....(ii)" "(lamda -a gt 0)`
Adding (i) and (ii)
`f(lambda) + f(b) le 2 f(a) `
`i.e., " " f(x) + f(b) le 10 " "` Hence range is `(-oo,10]`
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