Help calc1 problem

Question

h(x) = x − 2 cos(x), [0, 4𝜋] , find all local minimum by second derivative

Answer 1

The first step in finding minima with the second derivative is to first find the critical points, which is where h'(x) is equal to 0.

h'(x) = 1 + 2sin(x) | [0, 4π]

Solving for h'(x) = 0:

1 + 2sin(x) = 0

2sin(x) = -1

sin(x) = -1/2

x = 7π/6, 11π/6, 19π/6, 23π/6

The second step is to calculate h''(x) at each of these points. If h''(x) is positive, then the point is a local minimum.

h''(x) = 2cos(x)

2cos(7π/6) = -√3

2cos(11π/6) = √3

2cos(19π/6) = -√3

2cos(23π/6) = √3

Therefore, the two local minima are x = 11π/6 and x = 23π/6.

I hope this helps!