Question

Determine coefficients a and b such that p(x)=x2 ax b satisfies p(1)=5 and p′(1)=11.

Answer 1

$p(x)=x^2+ax+b$
$p'(x)=2x+a$

Given that $p'(1)=11$, you have

$p'(1)=2+a=11\implies a=9$

and given that $p(1)=5$, you have

$p(1)=1+9+b=5\implies b=-5$

So you get

$p(x)=x^2+9x-5$