Find the Inflection Points y=x^4-8x^3+16x^2

Problem

y=x4−8*x3+16*x2

Solution

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

d(y)/d(x)=4*x3−24*x2+32*x

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

d2(y)/(d(x)2)=12*x2−48*x+32

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

12*x2−48*x+32=0

4. Simplify the equation by dividing all terms by the greatest common divisor, which is 4

3*x2−12*x+8=0

5. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=3 b=−12 and c=8

x=(12±√(,(−12)2−4*(3)*(8)))/(2*(3))

6. Simplify the radical and solve for x

x=(12±√(,144−96))/6

x=(12±√(,48))/6

x=(12±4√(,3))/6

x=2±(2√(,3))/3

7. Determine the y-coordinates by substituting the x values back into the original function y=x4−8*x3+16*x2 which can be factored as y=(x2−4*x)2

y=((2±(2√(,3))/3)2−4*(2±(2√(,3))/3))2

y=(4±(8√(,3))/3+4/3−8∓(8√(,3))/3)2

y=(4+4/3−8)2

y=(−8/3)2

y=64/9

Final Answer

Inflection Points: *(2−(2√(,3))/3,64/9),(2+(2√(,3))/3,64/9)

Want more problems? Check here!