Answer:
D) I, II, and III
Step-by-step explanation:
Use implicit differentiation to find dy/dx.
x³ + y² − 12x + 16y = 28
3x² + 2y dy/dx − 12 + 16 dy/dx = 0
(2y + 16) dy/dx = 12 − 3x²
dy/dx = (12 − 3x²) / (2y + 16)
When the tangent line is horizontal, dy/dx = 0.
0 = (12 − 3x²) / (2y + 16)
0 = 12 − 3x²
0 = 4 − x²
x = ±2
When the tangent line is vertical, dy/dx is undefined, meaning the denominator is 0.
0 = 2y + 16
y = -8
Finally, at (-1, 1), the slope of the tangent line is:
dy/dx = (12 − 3(-1)²) / (2(1) + 16)
dy/dx = (12 − 3) / (2 + 16)
dy/dx = 1/2
All three options are true.
Graph: desmos.com/calculator/fhdromav4g
