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If H (x(0))=0 for some x=x(0)and (d)/(dx...

If H `(x_(0))=0` for some `x=x_(0)and (d)/(dx)H(x)gt2cxH(x)` for all `xgex_(0),` where `cgt0,` then prove that H (x) cannot be zero for any `xgtx_(0).`

A

H(x) = 0 has root for `x gt x_(0)`

B

H(x) = 0 has no root for `x gt x_(0)`

C

H(x) is a constant functio

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
2
Given that `(d)/(dx)H(x)gt2cxH(x)`
or `e^(-cx^(2))(d)/(dx)H(x)-e^(-cx^(2))2cxH(x)gt0`
or `(d)/(dx)H(x)e^(-cx^(2))gt0`
Thus `H(x)e^(-cx^(2))` is and increasing function
But `h(x_(0))gt0` for all `xgtx_(0)`
Hence H(x) connot be zero for any `xgtx_(0)`
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