Question

I just need the 3rd part!!!

(2 points) Let y be defined implicitly by the equation ln(2y)=5xy. Use implicit differentiation to find the first derivative of y with respect to x.

dy/dx= 5y^2/(1-5xy) (I ALREADY GOT THIS)


Use implicit differentiation to find the second derivative of y with respect to x.

d^2ydx^2= y^3(250xy-75)/((5xy-1)^3) (ALREADY GOT THIS)


Note: Your answer should only involve the variables x and y. You should simplify your answer as much as possible before entering it into WeBWorK.

Find the point on the curve where d^2ydx^2=0.

d2ydx2=0 at the point (x,y)=
. Include the parentheses in your answer.

Answer 1

Well, you want the 2nd derivative = 0
so
y^3(250xy-75)/((5xy-1)^3) = 0
y^3(250xy-75) = 0

y^3 = 0, but that would make ln(2y) as undefined, so y ≠ 0
or

250xy-75 = 0
xy = -75/250 = -3/10

but ln(2y)=5xy
xy = ln(2y)/5

ln(2y)/5 = -3/10
ln(2y) = - 3/2
2y = e^(-3/2)
y = .111565

xy = ln(2y)/5
x = ln(2y)/(5y) = -2.689