Find the Inflection Points f(x)=x^4-4x^3+5

Problem

ƒ(x)=x4−4*x3+5

Solution

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

d(ƒ(x))/d(x)=4*x3−12*x2

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

d2(ƒ(x))/(d(x)2)=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 for concavity changes by checking the sign of ƒ(x)″ in the intervals (−∞,0) (0,2) and (2,∞) Since ƒ(x)″ changes sign at both x=0 and x=2 both are inflection points.

ƒ″*(−1)=36>0

ƒ(1)″=−12<0

ƒ(3)″=36>0

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

ƒ(0)=(0)4−4*(0)3+5=5

ƒ(2)=(2)4−4*(2)3+5=16−32+5=−11

Final Answer

Inflection Points=(0,5),(2,−11)

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