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Let fandg be real valued functions defin...

Let `fandg` be real valued functions defined on interval `(-1,1)` such that `g''(x)` is constinous, `g(0)=0`, `g'(0)=0,g''(0)=0andf(x)=g(x)sinx`.
Statement I `underset(xrarr0)lim(g(x)cotx-g(0)cosecx)=f''(0)`
Statement II `f'(0)=g'(0)`

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(x to 0)lim [g(x)cot x-g(0)cosec x]`
`underset(x to 0)lim (g(x) cos x-g(0))/(sin x)" "((0)/(0)"form")`
`underset(x to 0)lim (g'(x) cos x-g(x)sin x)/(cos x)" "["Using L' Hospital's Rule"]`
`=0" "[therefore g'(0)=0 and g(0)=0]`
Now, f(x)=g(x) sin x
`Rightarrow f'(x)=g'(x) sin x+g(x) cos x.....(i)`
`Rightarrow f''(x)=g''(x) sin x+2g'(x) cos x-g(x)sin x`
`Rightarrow f''(0)=0`
`therefore underset(x to 0)lim [g(x)cot x-g(0) cosec x]=f''(0)" "[therefore f''(0)=0]`
So, statement-1 is true.
From (i), we have f'(0)=g(0)
So, statement-2 is true. But, statemetn-2 is not a correct explanation for statement-1.
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