Home
Class 12
MATHS
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]`

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 ) C in [a , b]`
ATQ `(f(b) - f(a))/(b - a) le 0`
ie `f(b) - f(a) le 0`
again using LMVT in `[a, lambda]`
`(f(lambda) - f(a))/(lambda - a) le 0`
`implies f(lambda) - f(a) le 0` `(lambda - a gt 0)`
adding (i) and (ii)
`f(lambda) + f (b) le 2 f (a)`
i.e `f(lambda) + f(b) le 10`. Hence range is `(- oo, 10]`
Promotional Banner

Similar Questions

Explore conceptually related problems

If f''(x) le0 "for all" x in (a,b) then f'(x)=0

Let the function f:[-7,0]rarr R be continuous on [-7,0] and differentiable on (-7,0) .If f(-7)=-3 and f'(x)<=2 , for all x in(-7,0) , then for all such functions f,f(-1)+f(0) lies in the interval

Tet f be continuous on [a,b] and differentiable on (a,b). If f(a)=a and f(b)=b, then

Let f be a differentiable function defined on R such that f(0) = -3 . If f'(x) le 5 for all x then

If f(x) is twice differentiable and continuous function in x in [a,b] also f'(x) gt 0 and f ''(x) lt 0 and c in (a,b) then (f(c) - f(a))/(f(b) - f(a)) is greater than

Let f be a differentiable real function defined on (a;b) and if f'(x)>0 for all x in(a;b) then f is increasing on (a;b) and if f'(x)<0 for all x in(a;b) then f(x) is decreasing on (a;b)

Let f:R rarr R be a twice differentiable function such that f(x+pi)=f(x) and f'(x)+f(x)>=0 for all x in R. show that f(x)>=0 for all x in R .