Consider second degree polynomial \(P(x)=a+bx+cx^{2}\) such that

\(P(4)=5, P′(4)=3\), and \(P′′ (4)=3.\)

\(P(4)=5P(4)=5\) gives us

\(a+4b+16c=5\).

Now differentiating P, we get \(P′=b+2cx\). Now putting \(x=4\), we get

\(b+8c=3\).

Now differentiating P′, we get \(P''=2cP\). Now putting \(x=4,\) we get

\(b+12=3\Rightarrow b=−9.\)

Now putting \(c=\frac{3}{2}\) and \(b= -9\ in\ {eq1}\) we get

\(a−36+48=5\Rightarrow a=−7.\)

So required second degree polynomial is \(P(x)=−7−9x+(\frac{3}{2})x2\)