find the y values

Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.

y = √3 x + 2 cos(x), 0 ≤ x < 2π

Find the y values by substituting the values from Step 2 into the original function. y1 corresponds to x1 and y2 corresponds to x2. List the points where the function has horizontal tangent lines. Submit your answers in terms of pi for π.

(x_1,y₁)= (___,___)

(x₂,y₂)= (___,___)

Answer 1

A horizontal tangent line means the rate of change of the function is 0. This also means that the first derivative is 0.

dy/dx = sqrt(3) - 2 sin(x). We let this equal 0 and get sin(x) = sqrt(3)/2. x must therefore be pi/3 or 2pi/3.

Plugging back into y, we get (pi/3, pi sqrt(3)/3 + 1) and (2pi/3, 2sqrt(3)pi/3 -1).

I hope this helps!