Find the Inflection Points -x^6+42x^5-42x+19

Problem

ƒ(x)=−x6+42*x5−42*x+19

Solution

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

d(ƒ(x))/d(x)=−6*x5+210*x4−42

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

d2(ƒ(x))/(d(x)2)=−30*x4+840*x3

3. Set the second derivative to zero to find potential inflection points.

−30*x4+840*x3=0

4. Factor the equation to solve for the variable x

−30*x3*(x−28)=0

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

x=0

x=28

6. Test for concavity changes by checking the sign of the second derivative on intervals around the critical values. Since the roots x=0 and x=28 have odd multiplicities (3 and 1 respectively), the sign of the second derivative changes at both points, confirming they are inflection points.

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

ƒ(0)=−(0)6+42*(0)5−42*(0)+19=19

ƒ(28)=−(28)6+42*(28)5−42*(28)+19=252,104,467

Final Answer

(0,19),(28,252104467)

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