Question

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

Answer 1

`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`.