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If a twice differentiable function f(x) ...

If a twice differentiable function f(x) on `(a,b)` and continuous on [a, b] is such that `f''(x)lt0` for all `x in (a,b)` then for any `c in (a,b),(f(c)-f(a))/(f(b)-f(c))gt`

A

`(b-c)/(c-a)`

B

`(c-a)/(b-c)`

C

`(b-c)(c-a)`

D

`(1)/((b-c)(c-a))`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `u in (a,c), v in (c,b)`
Then by LMCT on (a, c), (c, b), we have
`f'(u)=(f(c)-f(a))/(c-a),f'(v)=(f(b)-f(c))/(b-c)`
But `u lt v and f''(x)lt 0, AA x in (a,b)`
`"i.e. "f'(u)gtf'(v)`
`rArr" "(f(c)-(a))/(f(b)-f(c))gt(c-a)/(b-c)`
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