Question

Find the inflection points of f(x) = 4x^4 + 55x^3 - 21x^2 + 13. (Give your answered as a commas as a separated list) inflection points =

Answer 1

Okay, here we have this:

Considering the provided function, we are going to calculate the inflection points, so we obtain the following:

Let's remember that the inflection points are those where the derivative is equal to zero or is undefined and changes sign, so let's calculate it:

$\begin{gathered} f(x)=4x^4+55x^3-21x^2+13 \\ f^{\prime}^{\prime}(x)=\frac{d^2(4x^4+55x^3-21x^2+13)}{dx^2} \\ f^{\prime}^{\prime}(x)=48x^2+330x-42 \end{gathered}$

Now let's find the points where the second derivative is zero or undefined:

$\begin{gathered} 48x^2+330x-42=0 \\ x_{1,2}=\frac{-330\pm\sqrt{330^2-4\cdot48(-42)}}{2\cdot48} \\ x_{1,2}=\frac{-330\pm342}{96} \\ x_1=\frac{-330+342}{96},x_2=\frac{-330-342}{96} \\ x_1=\frac{12}{96},x_2=\frac{-672}{96} \\ x_1=\frac{1}{8},x_2=-7 \end{gathered}$

Now we will calculate the third derivative of the function and we will evaluate in these roots, if the result is different from zero then we have an inflection point:

$\begin{gathered} f^{\prime}^{\prime}^{\prime}(x)=\frac{d(f^{\prime}^{\prime}(x))}{dx} \\ =\frac{d(48x^2+330x-42)}{dx} \\ =96x+330 \end{gathered}$

And evaluating in the two roots:

$\begin{gathered} f^{\prime}^{\prime}^{\prime}(\frac{1}{8})=96(\frac{1}{8})+330 \\ =12+330 \\ =342 \\ f^{\prime}^{\prime}^{\prime}(-7)=96(-7)+330 \\ =-672+330 \\ =-342 \end{gathered}$

We observe that since the result is different from zero, then the two are inflection points, we substitute in the original function to find the y-coordinate of the points:

$\begin{gathered} f(\frac{1}{8})=4(\frac{1}{8})^4+55(\frac{1}{8})^3-21(\frac{1}{8})^2+13 \\ =\frac{13087}{1024} \\ f(-7)=4(-7)^4+55(-7)^3-21(-7)^2+13 \\ =-10277 \end{gathered}$

Finally we obtain that the inflection points are: (1/8, 13087/1024) and (-7, -10277).