Question

In the xy -plane, the point (0,-2) is on the curve C. If dy/dx = 4x/9y for the curve, which of the following statements is true?

a) At the point (0,-2) , the curve C has a relative minimum because dy/dx =0 and d²y/dx² >0.
b) At the point (0,-2) , the curve C has a relative minimum because dy/dx =0 and d²y/dx² <0.
c) At the point (0,-2) , the curve C has a relative maximum because dy/dx =0 and d²y/dx² >0.
d) At the point (0,-2) , the curve C has a relative maximum because dy/dx =0 and d²y/dx² <0.

Answer 1

The curve C has a relative maximum at the point (0,-2) because dy/dx = 0 and d²y/dx² < 0.

Step-by-step explanation:

To determine whether the curve C has a relative minimum or maximum at the point (0,-2), we need to analyze the first and second derivatives of the curve at that point.

Given that dy/dx = 4x/9y, we can substitute the x and y-values of the point (0,-2) into this expression to find the value of dy/dx. When x = 0 and y = -2, dy/dx = 0/(-18) = 0.

However, the question also asks about the value of the second derivative, d²y/dx², at the point (0,-2). Since d²y/dx² can be obtained by differentiating dy/dx with respect to x, we differentiate the expression 4x/9y with respect to x. The result is d²y/dx² = 4/9y.

Substituting y = -2 into this equation gives us d²y/dx² = 4/(-18) = -2/9. As this value is negative, we can conclude that at the point (0,-2), the curve C has a relative maximum because dy/dx = 0 and d²y/dx² < 0.