Math Problem Statement

If minus, 5, x, y, minus, y, squared, minus, 3, x, cubed, equals, 1−5xyy2−3x3=1 and it is known that start fraction, d, y, divided by, d, x, end fraction, equals, start fraction, 5, y, plus, 9, x, squared, divided by, minus, 5, x, minus, 2, y, end fraction, commadxdy​=−5x−2y5y+9x2​, find all coordinate points on the curve where x, equals, 2x=2 and the line tangent to the curve is vertical, or state that no such points exist.

Solution

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Vertical Tangent
Coordinate Geometry

Formulas

dy/dx = (5y + 9x^2) / (-5x - 2y)

Theorems

Implicit Differentiation
Condition for Vertical Tangent (dy/dx denominator = 0)

Suitable Grade Level

Grades 11-12