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AB Calc!!!
Let C be the curve defined by x2+y2=36. Consider all points (x,y) on curve C where y>0. Which of the following statements provides a justification for the concavity of the curve?

The curve is concave down because y′′=−2yx2<0.
The curve is concave up because y′′=2yx2>0.
The curve is concave down because y′′=−36y3<0.
The curve is concave up because y′′=36y3>0.

Answer :

Following are the calculation to the concavity of the curve:

Given equation:

[tex]\blod{x^2 + y^2 = 36}[/tex]

To find:

concavity of the curve=?

Solution:

[tex]\blod{x^2 + y^2 = 36}[/tex]

 Differentiating the above equation with the respect of 'x':  

[tex]\to 2x + 2y \frac{dy}{dx}=0\\\\ \to 2yy'=-2x\\\\ \to y'=-\frac{2x}{2 y} = -\frac{x}{y} \\\\[/tex]

Again Differentiating the value with the respect of 'x':  

[tex]\to y'' = -[ \frac{y-xy'}{y^2}]\\\\\to y''= -[ \frac{y-x(- \frac{x}{y})}{y^2}]\\\\\to y''= -[ \frac{y^2+ x^2}{y^3}]\\\\[/tex]

using the [tex]ep^{n}[/tex]of the curve:  

[tex]\to y''= -[ \frac{36}{y^3}]<0[/tex] so, the Concave down.

Therefore, the final answer is "third choice".

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brainly.com/question/2919483

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