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The number of polynomials p:R to R satis...

The number of polynomials `p:R to R` satisfying `p(0)=0,p(x) gtx^(2)` for all `x ne0`, and `p''(0)=(1)/(2)` is

A

0

B

1

C

more than 1, but finite

D

infinite

Text Solution

Verified by Experts

The correct Answer is:
A
Assume `g(x)=p(x)-x^(2)" " ` (g(x) is polynomial to differentiable function )
given `p(x) gt x^(2) implies p(x)-x^(2) gt o AA x ne0`
`implies g(x) gt o AA x ne 0`
and `g(0)=p(0)-0=0`
As `g(x) gt 0 AA x ne 0`
`impliesx=0` should be a minima
`:. g''(x)` should be `ge 0 " at " x=0`
Now `g'(x) =p'(x) - 2x`
and `g''(x)=p''(x)-2`
`=(1)/(2)-2`
`=-(3)/(2)` so, contradiction
`:.` No such polynomial exist
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