For what values of x in [0, 2pi] does the graph of Y= cos(x)/2+ sin(x)

Question

have a horizontal tangent? List the values of x below. Please explain?

Answer 1

The function y = cos(x)/2 + sin(x)

Differentiating on each side with respect to x.

y' = - sin(x)/2 + cos(x)

When the tangent line is horizontal, the slope is equal to 0.

That means, need to solve for y' = 0

- sin(x)/2 + cos(x) = 0

cos(x) = sin(x)/2

2 = sin(x)/cos(x)

tan(x) = 2

x = tan^-1(2)

x = 63.43

The genaral solution of tan(x) = tan(α) is x = nπ + α, where n is an integer.

x = nπ + 63.43

For n = 0, x = 0(180) + 6 3.43

x = 63.43⁰

For n = 2,x = 1(180) + 63.43

x = 243.43⁰

For n = 2, x = 2(180) + 63.43

x = 423.43⁰

The solutions on the interval [0, 2π] are x = 63.43⁰ and 243.43⁰.