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The differential equation dy/dx=2x(4-y)

I. produces a slope field with horizontal tangents at y = 4

II. produces a slope field with horizontal tangents at x = 0

III. produces a slope field with vertical tangents at x = 0 and y = 4



which is true?

i only

ii only

i and ii

iii only

Verified Answer :

apologiabiology

horizontal means dy/dx=0
vertical means that you are dividing by 0 or dy/dx is undefined



at x=0, dy/dx=0
at y=4, dy/dx=0
so the answer is I and II

answer is 3rd option

Answer:

The correct option is 3. Statement i and ii are correct.

Step-by-step explanation:

The given differential equation is

\(\frac{dy}{dx}=2x(4-y)\)

We know that \(\frac{dy}{dx}\) represents the slope of a function.

The slope of a vertical line is infinity and the slope of horizontal line is 0.

For horizontal tangents

\(\frac{dy}{dx}=0\)

\(2x(4-y)=0\)

Using zero product property equation each factor equal to 0.

\(2x=0\Rightarrow x=0\)

\(4-y=0\Rightarrow y=4\)

Therefore given differential equation the produces a slope field with horizontal tangents at y = 4 and at x=0. Statement i and ii are correct.

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