Find Where dy/dx is Equal to Zero y^2-3xy+x^2=7

Problem

y2−3*x*y+x2=7

Solution

1. Differentiate implicitly with respect to x treating y as a function of x and applying the product rule to the term −3*x*y

2*yd(y)/d(x)−3*(xd(y)/d(x)+y)+2*x=0

2. Distribute the constant through the parentheses.

2*yd(y)/d(x)−3*xd(y)/d(x)−3*y+2*x=0

3. Isolate the terms containing d(y)/d(x) on one side of the equation.

2*yd(y)/d(x)−3*xd(y)/d(x)=3*y−2*x

4. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(2*y−3*x)=3*y−2*x

5. Solve for the derivative by dividing both sides by (2*y−3*x)

d(y)/d(x)=(3*y−2*x)/(2*y−3*x)

6. Set the derivative to zero to find the condition where the slope is horizontal.

(3*y−2*x)/(2*y−3*x)=0

7. Solve for y by setting the numerator equal to zero.

3*y−2*x=0

y=2/3*x

8. Substitute y=2/3*x back into the original equation y2−3*x*y+x2=7 to find the specific points.

(2/3*x)2−3*x*(2/3*x)+x2=7

4/9*x2−2*x2+x2=7

−5/9*x2=7

9. Analyze the result to determine if real solutions exist. Since x2=−63/5 has no real solutions, there are no points on the curve where the derivative is zero.

Final Answer

d(y)/d(x)=0* at no real points

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