ANSWER
C and D
EXPLANATION
We want to identify the options that are correct for the end behavior of the function:
[tex]f(x)=-4x^6+6x^2-52[/tex]
First, let us test to see if the function is even.
A function is even if:
[tex]f(x)=f(-x)[/tex]
To find f(-x), substitute -x for x in the function:
[tex]\begin{gathered} f(-x)=-4(-x)^6+6(-x)^2-52 \\ \\ f(-x)=-4x^6+6x^2-52 \end{gathered}[/tex]
Since f(-x) is equal to f(x), we see that the function is even, hence, both ends of the graph go in the same direction.
The leading coefficient of a function is the coefficient of the term with the highest degree.
The leading coefficient of the given function is -4.
Since the leading coefficient is negative, it tends towards negative infinity as x increases, and so, the left end of the graph goes down.
Therefore, the options that are correct for the function are options C and D.
