How many point does the curve y^4-3x^3=2 have a vertical tangent line?

Question

How many points does the curve y^4-3x^3=2 have a vertical tangent line? details please. 

Answer 1

Hi Mariana,

 

An equation has a vertical tangent when dx/dy = 0. Taking the implicit derivative of your given curve,

 

y⁴ - 3x³ = 2

4y³ dy - 9x² dx = 0

4y³ dy = 9x² dx

4y³ / 9x² = dx/dy

 

Set it equal to 0 and solve,

4y³ / 9x² = 0

4y³ = 0

y = 0

 

Find the x-value(s) that correspond to this y-value,

y⁴ - 3x³ = 2

0⁴ - 3x³ = 2

x³ = -2/3

x = - ³√(2/3)

 

There's a single vertical tangent line at (- ³√(2/3), 0)