Identify all points at which the curve has A) a horizontal tangent and b) a vertical tangent? identify all poi

Identify all points at which the curve has
A) a horizontal tangent and
b) a vertical tangent?


x=cos2t
y=sin7t

You have the horizontal tangents figured out, so I'll leave you to it.

Vertical tangents are where the first derivative is infinite. The first derivatives cos7t and sin2t are bounded on the range [1,-1] so they are never infinite, therefore there are no vertical tangents.

Points of Inflection are found where the second derivative is 0

y" = -49sin7t = 0

Which is true for 7t = kπ, k= ±0,±1,±2,...

or t = kπ/7, k = ±0,±1,±2,...

*SIDE NOTE* Since these are simple sinusoidal functions, Points of Inflection (y"=0) happen to coincide with points where y=0

You have done it for the horizontal tangent .
For the second part, there is no vertical tangent for a sinusoidal curve.