# Find the values?

## Find the values of a and b so that the function f(x) = ${x}^{2}$ + $a x$ + b has the tangent line 2y − 4x = 2 at the point (2, 5).

Mar 19, 2018

$a = - 2 , b = 5$

#### Explanation:

Th generic tangent line to $f \left(x\right)$ at ${x}_{0}$ is

$y = f \left({x}_{0}\right) + f ' \left({x}_{0}\right) \left(x - {x}_{0}\right)$

here

$f \left({x}_{0}\right) = {x}_{0}^{2} + a {x}_{0} + b$ and
$f ' \left({x}_{0}\right) = 2 {x}_{0} + a$ then comparing

$y = {x}_{0}^{2} + a {x}_{0} + b + \left(2 {x}_{0} + a\right) \left(x - {x}_{0}\right)$ at ${x}_{0} = 2$

$y = 4 + 2 a + b + \left(4 + a\right) \left(x - 2\right) = b - 4 + \left(a + 4\right) x \equiv y = 2 x + 1$

so

$\left\{\begin{matrix}b - 4 = 1 \\ a + 4 = 2\end{matrix}\right.$

and solving

$a = - 2 , b = 5$