Find the Inflection Points y=6x^4+8x^3

Problem

y=6*x4+8*x3

Solution

1. Find the first derivative by applying the power rule to each term of the function.

d(y)/d(x)=24*x3+24*x2

2. Find the second derivative by differentiating the first derivative with respect to x

d2(y)/(d(x)2)=72*x2+48*x

3. Set the second derivative to zero to find potential inflection points where the concavity might change.

72*x2+48*x=0

4. Factor the equation to solve for the critical values of x

24*x*(3*x+2)=0

5. Solve for x by setting each factor equal to zero.

x=0

x=−2/3

6. Verify the inflection points by checking if the second derivative changes sign around these values. Since 72*x2+48*x is a parabola opening upward with roots at 0 and −2/3 the sign changes at both points.

7. Calculate the y-coordinates by substituting the x values back into the original function y=6*x4+8*x3

y(0)=6*(0)4+8*(0)3=0

y*(−2/3)=6*(−2/3)4+8*(−2/3)3=6*(16/81)+8*(−8/27)=32/27−64/27=−32/27

Final Answer

Inflection Points:(0,0),(−2/3,−32/27)

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