Q. 383.5( 4 Votes )
Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?
Answer :
As f(0) = f(1) and f is differentiable, hence by Rolles theorem:
for some c 
[0,1]
let us now apply LMVT (as function is twice differentiable) for point c and x 
[0,1],
hence,
f ”(d)
⇒ 
f ”(d)
⇒ 
f ”(d)
A given that | f ”(d)| <=1 for x 
[0,1]
⇒ 
⇒ 
Now both x and c lie in [0,1], hence 
[0,1]
Rate this question :
Different kind of mappings58 mins
Rapid Quiz challenge - Cell Structure & Cell Division52 mins
Functions - 0152 mins
Functions - 1047 mins
Functions - 0558 mins
Functions - 0748 mins
Functions - 0253 mins
Functions - 0361 mins
Functions - 0840 mins
Functions - 1252 mins
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 6 + 12x + 3x2 – 2x3
RD Sharma - Volume 1Find the intervals in which the following functions are increasing or decreasing.
f(x) = x4 – 4x
RD Sharma - Volume 1Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 – 9x2 + 12x – 5
RD Sharma - Volume 1Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 – 24x + 107
RD Sharma - Volume 1Find the intervals in which the following functions are increasing or decreasing.
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x3 – 24x + 7
RD Sharma - Volume 1
