Respuesta :
Given the function:
[tex]f(x)=-x^4+24x^2[/tex]You need to remember that the Inflection Points of a function are where it changes its concavity.
By definition, a function can have more than one inflection point:
1. You need to find:
[tex]f^{\prime}(x)[/tex]Remember the following Derivatives Power Rule:
[tex]\frac{d}{dx}(x^n)=nx^{n-1}[/tex]You get:
[tex]f^{\prime}(x)=-4(x^{4-1})+(2)(24)(x^{2-1})[/tex][tex]f^{\prime}(x)=-4x^3+48x[/tex]2. Find the second derivative by derivating the first derivative. This is:
[tex]f^{^{\prime}^{\prime}}(x)=(-4)(3)x^{3-1}+48[/tex][tex]f^{^{\prime\prime}}(x)=-12x^2+48[/tex]3. Find the third derivative by derivating the second derivative:
[tex]f^{^{\prime^{\prime}^{\prime}}}(x)=(-12)(2)x^[/tex][tex]f^{^{\prime^{\prime}^{\prime}}}(x)=-24x[/tex]4.Find the roots of the second derivative:
- Set up that:
[tex]-12x^2+48=0[/tex]- And solve for "x":
[tex]\begin{gathered} -12x^2=-48 \\ \\ x=\sqrt{\frac{-48}{-12}} \\ \\ x_1=2 \\ x_2=-2 \end{gathered}[/tex]5. Substitute each root into the third derivative and evaluate:
[tex]f^{^{\prime^{\prime}^{\prime}}}(2)=-24(2)=-48[/tex]It is less than zero, therefore there is an inflection point in that x-value.
[tex]f^{^{\prime^{\prime\prime}}}(-2)=-24(-2)=48[/tex]It is positive output value, therefore there is an inflection point in that x-value.
6. Substitute those x-values into the original function and evaluate:
[tex]f(2)=-(2)^4+24(2)^2=80[/tex][tex]f(-2)=-(-2)^4+24(-2)^2=80[/tex]Therefore, the inflections points are:
[tex]\begin{gathered} (-2,80) \\ (2,80) \end{gathered}[/tex]Hence, the answer is:
There are two inflection points:
[tex]\begin{gathered} (-2,80) \\ (2,80) \end{gathered}[/tex]