Where the function f(x) = cotx + 4x is horizontal?

Find the values of x between 0 and 2pi where the function f(x) = cotx + 4x is horizontal.

1 Answer
Feb 3, 2018

#x=+-pi/6, +-(5pi)/6#

Explanation:

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In order to find the slope of a tangent line to a curve at a given point, we have to take the first derivative of the function of the curve and plug in the coordinates of the given point.

If the function #f(x)=y# its derivative will be a function of #x# and plugging the #x#-coordinate of the given point will give us the slope of the tangent at that point.

But in this case we want to find where the function is horizontal. This means we have to find the point(s) where the tangent to the curve is horizontal.

That means the slope of the tangent has to be #=0#.

#f(x)=cotx+4x#

#f'(x)=-csc^2x+4#

#m=-csc^2x+4=0#

#csc^2x=4#

#cscx=+-2#

#1/sinx=+-2#

#sinx=+-1/2#

Between #0# and #2pi#:

#x=+-pi/6, +-(5pi)/6#