Question

Find the values?

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

Answer 1

`a = -2, b = 5`

Explanation:

Th generic tangent line to `f(x)` at `x_0` is

`y = f(x_0) + f'(x_0)(x-x_0)`

here

`f(x_0) = x_0^2+ax_0 + b` and
`f'(x_0) = 2x_0+a` then comparing

`y = x_0^2+ax_0 + b+(2x_0+a)(x-x_0)` at `x_0 = 2`

`y = 4+2a+b+(4+a)(x-2) = b-4+(a+4)xequiv y =2x+1`

so

`{(b-4=1),(a+4=2):}`

and solving

`a = -2, b = 5`