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If f(x) is a twice differentiable functi...

If f(x) is a twice differentiable function and given that f(1) = 1, f(2) = 4, f(3) = 9, then

A

`f''(x) = 2, AA x in (R )`

B

` f'(x) = 5 = f''(x)," for some " x in (1, 3)`

C

there exists atleast one ` x in (1, 3) ` such that f''(x) = 2

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
C
Let, ` g(x) = f(x) - x^(2) `
` rArr` g(x) has atleast 3 real roots which are x = 1, 2, 3 [by mean value theorem]
` rArr` g'(x) has atleast 2 real roots in ` x in (1, 3)`
`rArr ` g''(x) has atleast 1 real roots in ` x in (1, 3) `
`rArr` f''(x) = 2, for atleast one root in ` x in (1, 3) `
` rArr` f''(x) = 2, for atleast one root in ` x in (1, 3) `
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