The normal to a given curve is parallel to x-axis if

Concept:

Normal is perpendicular to tangent, such that  mt × mn = -1

mt = dy/dx.

Calculation:

Let us consider a function f(x) , such that slope of tangent be m_t and slope of normal be m_n

Now , we know that, normal is perpendicular to tangent, such that  mt × mn = -1

mn × dy/dx = -1

⇒ m_n = - dx/dy

Now, as per the question, the normal is parallel to the x-axis, so the slope of normal is equal to the slope of the x-axis (i.e. zero)

∴ mn = 0

⇒ - dx/dy = 0

⇒ dx/dy = 0

∴ The normal to a given curve is parallel to x-axis if  dx/dy = 0