Find the Horizontal Tangent Line f(x)=x^3-4x^2

Problem

ƒ(x)=x3−4*x2

Solution

1. Find the derivative of the function to determine the slope of the tangent line at any point x

d(ƒ(x))/d(x)=3*x2−8*x

2. Set the derivative to zero because horizontal tangent lines occur where the slope is zero.

3*x2−8*x=0

3. Factor the equation to solve for the xvalues where the slope is zero.

x*(3*x−8)=0

4. Solve for x by setting each factor to zero.

x=0

x=8/3

5. Find the y-coordinates by substituting the xvalues back into the original function ƒ(x)

ƒ(0)=(0)3−4*(0)2=0

ƒ*(8/3)=(8/3)3−4*(8/3)2=512/27−256/9=−256/27

6. Write the equations of the horizontal lines using the ycoordinates found.

y=0

y=−256/27

Final Answer

y=0,y=−256/27

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