For what values of #x# does #f(x) = 4x + 8 sin x# have a horizontal tangent line?

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Answer 1 · 2018-08-04T04:22:11

# x=2kpi+-2pi/3, k in ZZ#.

Recall that the slope of tangent line (tgt.) is given by #f'(x)#.

So, if the tgt. is horizontal, its slope, i.e., #f'(x)=0.#

Now, #f(x)=4x+8sinx rArr f'(x)=4+8cosx#.

#:. f'(x)=0 rArr 4+8cosx=0 rArr cosx=-4/8=-1/2#.

But, #costheta=cosalpha rArr theta=2kpi+-alpha, k in ZZ#.

Therefore, #cosx=-1/2=cos(pi-pi/3)=cos(2pi/3),#

# rArr x=2kpi+-2pi/3, k in ZZ#.