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

1 Answer
Aug 4, 2018

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

Explanation:

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.