Find the quadratic polynomial ax^2+bx+c which best fits the function f (x) = 8^x at x=0, in the sense that g (0) = f (0), and f' (0) = g' (0), and f'' (0) = g'' (0).

g (x) = ?

F (0) = 8^0 = 1

f ' (x) = 8^x * ln8; x = 0 = > f ' (0) = ln8

f '' (x) = 8^x (ln8) ^2; x = 0=> f '' (0) = (ln8) ^2

g (0) = a (0) ^2 + b (0) + c = f (0) = 1 = > c = 1

g ' (x) = 2ax + b = > g ' (0) = 2a (0) + b = f ' (0) = ln8 = > b = ln (8)

g'' (x) = 2a = > g'' (0) = 2a = f'' (0) = (ln8) ^2 = > a = (ln8) ^2 / 2

=> g (x) = [ (ln8) ^2 / 2] x^2 + [ln (8) ]x + 1