The function y= x^4 + b*x^2 + 8x+1. It has a horizontal tangent and a point of inflection for the same value of x.
The slope of a tangent at any point x is given by the value of y' at that point. As the tangent is horizontal at the required value of x, y' = 0
=> 4x^3 + 2bx + 8 = 0
=> 2x^3 + bx + 4 = 0 ...(1)
Also, the function has a point of inflection for the same value of x, y'' = 0
=> 12x^2 + 2b = 0
=> 6x^2 + b = 0 ...(2)
From (1) and (2) one real root of b is obtained and it is equal to b = -6.
The required value of b = -6.