Published by:
CGP EDU Academic Team
Published on: August 13, 2026
For every twice differentiable function f: R 
[-2, 2] with (f(0)) 2 + (f'(O)) 2 = 85, which of the following statement(s) is (are) TRUE ?
A
There exist r, s 
R, where r < s, such that f is one-one on the open interval (r, s)
R, where r < s, such that f is one-one on the open interval (r, s)
B
There exists x 0 
(-4, 0) such that |f'(x 0 )| 
1
(-4, 0) such that |f'(x 0 )| 1
C
D
There exist 
(-4, 4) such that f( 
) + f"( 
) = 0 and f'( 
) 
0
(-4, 4) such that f( ) + f"( ) = 0 and f'( ) 0
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
(a, b, d)
L.M.V.T. in [-4, 0]
If f(x) periodic then 
Similarly 
for some x 1 
(0, 4)
g(x) = (f(x)) 2 + (f’(x)) 2
g(x 0 ) 
5, g(x l ) 
5
g(0) = 85 it has a local maximum having value 
85
Say 
g'( 
) = 0, g"( 
)<0
2f( 
)f’( 
) + 2f’( 
)f”( 
) = 0
f'( 
)(f( 
) + f”( 
)) = 0
as f’ 
0
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