Find the Inflection Points x^4-4x^3+10

Problem

x4−4*x3+10

Solution

1. Find the first derivative by applying the power rule to each term of the function ƒ(x)=x4−4*x3+10

(d(x4)−4*x3+10)/d(x)=4*x3−12*x2

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

(d(4)*x3−12*x2)/d(x)=12*x2−24*x

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

12*x2−24*x=0

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

12*x*(x−2)=0

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

x=0

x=2

6. Test the intervals around x=0 and x=2 in the second derivative ƒ(x)″=12*x*(x−2) to confirm a change in sign (concavity).
   For x<0 ƒ″*(−1)=36>0 (concave up).
   For 0<x<2 ƒ(1)″=−12<0 (concave down).
   For x>2 ƒ(3)″=36>0 (concave up).
   Since the sign changes at both x=0 and x=2 both are xcoordinates of inflection points.

7. Calculate the y-coordinates by substituting the xvalues back into the original function ƒ(x)=x4−4*x3+10

ƒ(0)=(0)4−4*(0)3+10=10

ƒ(2)=(2)4−4*(2)3+10=16−32+10=−6

Final Answer

Inflection Points=(0,10),(2,−6)

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