Answer:
D) dy/dx > 0 and d²y/dx² > 0
Explanation:
Use implicit differentiation to find dy/dx and d²y/dx².
x²y³ = 576
x² (3y² dy/dx) + (2x) y³ = 0
3x²y² dy/dx = -2xy³
3x dy/dx = -2y
dy/dx = -2y / (3x)
d²y/dx² = [ (3x) (-2 dy/dx) − (-2y) (3) ] / (3x)²
d²y/dx² = (-6x dy/dx + 6y) / (9x²)
d²y/dx² = (-6x (-2y / (3x)) + 6y) / (9x²)
d²y/dx² = (4y + 6y) / (9x²)
d²y/dx² = 10y / (9x²)
Evaluating each at (-3, 4):
dy/dx = -2(4) / (3(-3))
dy/dx = 8/9
d²y/dx² = 10(4) / (9(-3)²)
d²y/dx² = 40/81
Both are positive.