Question

Find a second degree polynomial P such that P(2)=5, P'(2)=3, and P''(2)=2

Answer 1

Let
`P(x)=ax^2+bx+c`. You have derivatives
`P'(x)=2ax+b` and
`P''(x)=2a`. Then


`\begin{cases}P(2)=4a+2b+c=5\\P'(2)=4a+b=3\\P''(2)=2a=2\end{cases}`

From the last equation, you have
`a=1`. Plug this into the second equation to get
`4+b=3\implies b=-1`. Plug these into the first to get
`4-2+c=5\implies c=3`.

So the polynomial is


`P(x)=x^2-x+3`