Let f(x)=(x^2)+x+14. What is the value of f(x) for which the tangent line to the graph of y=f(x) is parallel to the x-axis?

Answer 1

In other words, where is the derivative zero or where is the vertex.

f' = 2x + 1
0 = 2 x + 1
x = -1/2

check by completing the square
x^2 + x = - 14
x^2 + x + 1/4 = -14 + 1/4
(x+1/2)^2 = - 13 3/4
sure enough, vertex at x = -1/2

Answer 2

f(-1/2) = 1/4 -1/2 + 14
= 13 1/4
remarkable :)