A. For what values of x does f (x )equals 3 x plus 3 sine x have a horizontal tangent line? b. For what values of x does f (x )equals 3 x plus 3 sine x have a slope of 3?

(a)Therefore the value of x=

(b) Therefore the value of x

Step-by-step explanation:

Horizontal tangent line: The first order derivative of a function gives the slope of the tangent of the function. The slope of horizontal line is zero.If the slope of tangent line is zero then the tangent line is called horizontal tangent line.

(a)

Given function is,

Differential with respect to x

For horizontal tangent line, f'(x)=0

3+ 3 cos x= 0

⇒3 cos x=-3

⇒cos x=-1

⇒x = 180°

Therefore the value of x=

(b)

Given that, the slope is 3.

Then,f'(x)=3

3+ 3 cos x= 3

⇒3 cos x= 3-3

⇒cos x=0

⇒x = 90°

Therefore the value of x