Question

Find the inflection points of f(x) = 2x^4 + 26x^3 - 42x^2 + 7. (Give your answers as a comma separated list)inflection points =

Answer 1

Given: The function below

`f(x)=2x^4+26x^3-42x^2+7`

To Determine: The interval of the inflection points

Solution:

The inflection points is the point where the second derivative of the function is equal to zero

Step 1: Determine the first derivative

`\begin{gathered} f(x)=2x^4+26x^3-42x^2+7 \\ f^(\prime)(x)=8x^3+76x^2-84x \end{gathered}`

Step 2: Determine the second derivative

`\begin{gathered} f^(\prime)(x)=8x^3+78x^2-84x \\ f^(\prime)^(\prime)(x)=24x^2+156x-84 \end{gathered}`

Step 3: Equate the second derivative to zero

`24x^2+156x-84=0`
`\begin{gathered} 24x^2-12x+168x-84=0 \\ 12x(2x-1)+84(2x-1)=0 \\ (2x-1)(12x+84)=0 \\ 2x-1=0,or12x+84=0 \\ 2x=1,or12x=-84 \\ x=(1)/(2),or\text{ }x=-(84)/(12) \\ x=(1)/(2),or\text{ }x=-7 \end{gathered}`