Find the Inflection Points x^4-6x^2+5

Problem

x4−6*x2+5

Solution

1. Find the first derivative by applying the power rule to each term of the function ƒ(x)=x4−6*x2+5

(d(x4)−6*x2+5)/d(x)=4*x3−12*x

2. Find the second derivative by differentiating the first derivative ƒ(x)′=4*x3−12*x

(d(4)*x3−12*x)/d(x)=12*x2−12

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

12*x2−12=0

4. Solve for x by factoring the quadratic equation.

12*(x2−1)=0

12*(x−1)*(x+1)=0

x=1,x=−1

5. Verify the concavity change by testing intervals around the critical values in ƒ(x)″ For x<−1 ƒ″*(−2)=36>0 (concave up). For −1<x<1 ƒ(0)″=−12<0 (concave down). For x>1 ƒ(2)″=36>0 (concave up). Since the sign changes at both points, they are inflection points.

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

ƒ(1)=(1)4−6*(1)2+5=0

ƒ*(−1)=(−1)4−6*(−1)2+5=0

Final Answer

Inflection Points=(1,0),(−1,0)

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