## Calculus 8th Edition

$$P(x)=x^2-x+3$$
Let the second-degree polynomial be $P(x)=ax^2+bx+c$, where $a\neq 0$. Differentiating $P$ with respect to $x$, $$P'(x)=2ax+b$$Again differentiating $P$, $$P''(x)=2a$$It is given that $P(2)=5, P'(2)=3$ & $P''(2)=2$. Thus, $$P''(2)=2a=2\implies a=1.$$Now, $$P'(2)=2a\times 2+b=3$$ $$4(1)+b=3\implies b=-1$$Now, $$P(2)=a(2)^2+b(2)+c=(1)(2)^2+(-1)(2)+c=5$$ $$4-2+c=5\implies c=3$$Thus, the required polynomial is, $$P(x)=x^2-x+3.$$