Find the Horizontal Tangent Line y=x^4-3x^2+2

Problem

y=x4−3*x2+2

Solution

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

d(y)/d(x)=4*x3−6*x

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

4*x3−6*x=0

3. Factor the equation to solve for the xcoordinates of the points where the tangent is horizontal.

2*x*(2*x2−3)=0

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

x=0

2*x2=3⇒x2=3/2⇒x=±√(,3/2)

5. Substitute the x values back into the original equation y=x4−3*x2+2 to find the corresponding ycoordinates.

For *x=0:y=(0)4−3*(0)2+2=2

For *x=±√(,3/2):y=(3/2)2−3*(3/2)+2=9/4−9/2+2=−1/4

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

Final Answer

y=2,y=−1/4

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