Find the Inflection Points f(x)=x^4-2x^3

Problem

ƒ(x)=x^4−2*x^3

Solution

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

d(ƒ(x))/d(x)=4*x^3−6*x^2

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

d^2(ƒ(x))/(d(x)^2)=12*x^2−12*x

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

12*x^2−12*x=0

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

12*x*(x−1)=0

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

x=0

x=1

6. Test for concavity changes by checking the sign of ƒ(x)^″ in the intervals (−∞,0) (0,1) and (1,∞) Since ƒ(x)^″ is a parabola opening upward with roots at 0 and 1 it is positive on (−∞,0) negative on (0,1) and positive on (1,∞) Because the sign changes at both x=0 and x=1 both are inflection points.

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

ƒ(0)=(0)^4−2*(0)^3=0

ƒ(1)=(1)^4−2*(1)^3=−1

Final Answer

Inflection Points: *(0,0),(1,−1)

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