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  4. The number of polynomials P:R → R satisfying P(0)=0, P(x)> x2 for all x ≠ 0 and p''(0)=(1/2) is

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Q. The number of polynomials $P:R \to R$ satisfying $P(0)=0, P(x)>\,x^{2}$ for all $x \ne 0$ and $p''(0)=\frac{1}{2}$ is

KVPYKVPY 2018

Solution:

We have,
$p(x)>\,x^{2}, p(0) =0, p''(0)=\frac{1}{2}$
Let $g(x)=p(x)-x^{2}$
$g(x)>\, 0, \forall x \ne 0$
and $g(0)=P(0)-0=0$
$\Rightarrow x=0$ should be minima
$\because g''(x)$ should be $\ge\,0$ at $x=0$
Now, $g'(x)=p'(x)-2x$
$g''(x)^{2}=p''(x)-2$
$g''(0)=P''(0) -2=\frac{1}{2}-2=-\frac{3}{2}$
But $g''(0) \ne 0$
$\because$ No polynomial exists