Find the Horizontal Tangent Line y = square root of 3x+2cos(x)

Problem

y=√(,3)*x+2*cos(x)

Solution

1. Identify the condition for a horizontal tangent line, which occurs when the derivative of the function is equal to zero.

y′=0

2. Differentiate the function with respect to x using the sum rule and basic trigonometric derivatives.

d(y)/d(x)=√(,3)−2*sin(x)

3. Set the derivative equal to zero to find the critical points.

√(,3)−2*sin(x)=0

4. Isolate the sine function.

2*sin(x)=√(,3)

sin(x)=√(,3)/2

5. Solve for x within the standard interval [0,2*π)

x=π/3

x=(2*π)/3

6. Substitute the x values back into the original function to find the corresponding y coordinates.

y(π/3)=√(,3)*(π/3)+2*cos(π/3)

y(π/3)=(π√(,3))/3+1

y((2*π)/3)=√(,3)*((2*π)/3)+2*cos((2*π)/3)

y((2*π)/3)=(2*π√(,3))/3−1

7. Write the equations of the horizontal tangent lines, which are in the form y=c

Final Answer

y=(π√(,3))/3+1,y=(2*π√(,3))/3−1

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