A curve represented parametrically by the equation x = t 3 − 4 t 2 − 3 t and y = 2 t 2 + 3 t − 5 where t ∈ R . If H denotes the number of point(s) on the curve where the tangent is horizontal and V is

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A curve represented parametrically by the equation x = t 3 − 4 t 2 − 3 t and y = 2 t 2 + 3 t − 5 where t ∈ R . If H denotes the number of point(s) on the curve where the tangent is horizontal and V is the number of point(s) where the tangent is vertical then

TestHub · Accepted Answer

d t d y ​ = 4 t + 3 , d t d x ​ = 3 t 2 − 8 t − 3 . Tangents are horizontal if d x d y ​ = 0 ⇒ d t d y ​ = 0 , d t d x ​  = 0 : 4 t + 3 = 0 ⇒ t = 4 − 3 ​ (one value, and d t d x ​  = 0 there) Tangents are vertical if d t d x ​ = 0 , d t d y ​  = 0 : 3 t 2 − 8 t − 3 = 0 ⇒ t = 3 , 3 − 1 ​ (two values, d t d y ​  = 0 at both) ∴ H = 1 , V = 2

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