Explanation:
The critical points are the points where the derivative of a function is zero or is not defined.
In this problem we have a function that's a polynomial and so, it's derivative is also a polynomial. Therefore, it has no value where it's not defined. However, there will be points where it's zero.
Let's find the derivarive first:
[tex]\frac{dg(x)}{dx}=g^{\prime}(x)=15x^4-15x^2[/tex]
We can rewrite it as:
[tex]g^{\prime}(x)=15x^2(x^2-1)[/tex]
So the zeros are now very easy to find. We can see that if x = 0, since we have x² multiplying, the whole function is zero. This is one critical point.
Then the function is also zero when (x²-1)=0:
[tex]\begin{gathered} x^2-1=0 \\ x^2=1 \\ x=\pm\sqrt[]{1} \\ x=\pm1 \end{gathered}[/tex]
The other two critical points are 1 and -1.
Answer:
There are 3 critical points within the domain: -1, 0, 1
