Answer
The equation of the second-order polynomial $P$ is $$P=x^2-x+3$$
Work Step by Step
A second-order polynomial $P$ would have the following equation form: $$P=ax^2+bx+c$$
Therefore,
- First derivative: $P'=a\frac{d}{dx}(x^2)+b\frac{d}{dx}(x)+\frac{d}{dx}(c)=2ax+b$
- Second derivative: $P''=2a\frac{d}{dx}(x)+\frac{d}{dx}(b)=2a$
We know that $P''(2)=2$. So, $$2a=2$$ $$a=1$$
We also know that $P'(2)=3$. So, $$2a\times2+b=3$$ $$2\times1\times2+b=3$$ $$b=-1$$
Finally, we know that $P(2)=5$. So, $$a\times(2^2)+b\times2+c=5$$ $$1\times4+(-1)\times2+c=5$$ $$c=3$$
Therefore, the equation of the second-order polynomial $P$ is $$P=x^2-x+3$$
