Sagot :
Given: The function below
[tex]f(x)=2x^4+26x^3-42x^2+7[/tex]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
[tex]\begin{gathered} f(x)=2x^4+26x^3-42x^2+7 \\ f^{\prime}(x)=8x^3+76x^2-84x \end{gathered}[/tex]Step 2: Determine the second derivative
[tex]\begin{gathered} f^{\prime}(x)=8x^3+78x^2-84x \\ f^{\prime}^{\prime}(x)=24x^2+156x-84 \end{gathered}[/tex]Step 3: Equate the second derivative to zero
[tex]24x^2+156x-84=0[/tex][tex]\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=\frac{1}{2},or\text{ }x=-\frac{84}{12} \\ x=\frac{1}{2},or\text{ }x=-7 \end{gathered}[/tex][tex]undefined[/tex]