For what values of x does the graph of f(x) = x + 2 sin x have a horizontal tangent?

Question

Must only be the smallest positive value of x

Answer 1

Horizontal tangents to the graph of f(x) are tangent lines that have slope = 0, so to find these "critical points" for f, we take its derivative, set f^'(x) = 0, and solve:

f(x) = x + 2 sinx

f^'(x) = 1 + 2cosx = 0

cosx = - 1/2

x = 2π/3 or 4π/3

Answer 2

Find derivative of function. f''(x)=1+2cosx. Set this equal to zero and solve for x. cox=-1/2. General solution of x will be 2/3pi + (2pi)k and 4/3pi + (2pik), for all integer k.