Find the point of inflection of the graph of the function. Describe the concavity. f(x) = x + 4 cos x, [0, 2π]

Answer 1

f'(x) = 1 - 4sinx; f''(x) = - 4cosx = 0, cosx = 0 and we have 2 points x = π/2 and x = 3π/2

So we have inflection point (π/2, π/2), (3π/2, 3π/2).

(0, π/2) f''(x ) < 0 so f(x) graph concave down,

(π/2, 3π/2) f''(x) > 0 so graph concave up

(3π/2, 2π) f''(x) < 0 so graph f(x) concave down