## Calculus: Early Transcendentals 8th Edition

The equation of the second-order polynomial $P$ is $$P=x^2-x+3$$
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 $$P=x^2-x+3$$