Find the Horizontal Tangent Line y^3+xy-y=8x^4

Problem

y3+x*y−y=8*x4

Solution

1. Differentiate implicitly with respect to x to find the derivative d(y)/d(x)

d(y3)/d(x)+(d(x)*y)/d(x)−d(y)/d(x)=(d(8)*x4)/d(x)

2. Apply the power rule and product rule to the terms.

3*y2d(y)/d(x)+(y+xd(y)/d(x))−d(y)/d(x)=32*x3

3. Isolate the derivative d(y)/d(x) by factoring out the common terms.

d(y)/d(x)*(3*y2+x−1)=32*x3−y

d(y)/d(x)=(32*x3−y)/(3*y2+x−1)

4. Set the derivative to zero to find the condition for a horizontal tangent line.

(32*x3−y)/(3*y2+x−1)=0

32*x3−y=0

y=32*x3

5. Substitute the expression for y back into the original equation to solve for x

(32*x3)3+x*(32*x3)−(32*x3)=8*x4

32768*x9+32*x4−32*x3=8*x4

32768*x9+24*x4−32*x3=0

6. Solve for x by factoring out 8*x3

8*x3*(4096*x6+3*x−4)=0

x=0

7. Find the y-coordinate by substituting x=0 into the original equation.

y3+(0)*y−y=8*(0)4

y3−y=0

y*(y2−1)=0

y=0,y=1,y=−1

8. Verify the denominator of the derivative is not zero at these points. At (0,0) the denominator 3*(0)2+0−1=−1≠0 At (0,1) 3*(1)2+0−1=2≠0 At (0,−1) 3*(−1)2+0−1=2≠0

9. Identify the horizontal tangent lines using the valid y values.

Final Answer

y=0,y=1,y=−1

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