To analyze the function [tex]\( f(x) = 2(x-4)^4 \)[/tex] and determine which statements are true, let's go through each statement one by one:
### Statement A: The left end of the graph of the function goes up, and the right end goes down.
For this, we need to consider the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( +\infty \)[/tex]. The function is a polynomial function of the form [tex]\( 2(x-4)^4 \)[/tex]. Since the leading term is positive and of even degree, both ends of the graph will rise to positive infinity. Therefore, this statement is false.
### Statement B: It has 4 zeros and at most 3 relative maximums or minimums.
To have 4 zeros, the function must intersect the x-axis at 4 different points. However, [tex]\( 2(x-4)^4 = 0 \)[/tex] only when [tex]\( x = 4 \)[/tex], and thus the function has exactly one zero. Therefore, this statement is false. The function could also have relative extrema, but not according to the logic provided in the statement.
### Statement C: It is a translation of the parent function 4 units to the left.
If we consider the parent function [tex]\( g(x) = 2x^4 \)[/tex], translating it 4 units to the left would yield [tex]\( 2(x+4)^4 \)[/tex] instead of [tex]\( 2(x-4)^4 \)[/tex]. Therefore, this statement is false.
### Statement D: It is a translation of the parent function 4 units to the right.
If we consider the parent function [tex]\( g(x) = 2x^4 \)[/tex], translating it 4 units to the right would yield [tex]\( 2(x-4)^4 \)[/tex], which matches our original function. Therefore, this statement is true.
### Statement E: Both ends of the graph of the function go up.
Since the leading term is positive and the degree of the polynomial is even, both ends of the function will tend toward positive infinity. Therefore, this statement is true.
### Statement F: It has 3 zeros and at most 4 relative maximums or minimums.
As discussed with Statement B, [tex]\( 2(x-4)^4 = 0 \)[/tex] only when [tex]\( x = 4 \)[/tex], giving the function only one zero. Therefore, this statement is false.
#### Conclusion:
True Statements:
- D. It is a translation of the parent function 4 units to the right.
- E. Both ends of the graph of the function go up.
The correct statements describing the function [tex]\( f(x)=2(x-4)^4 \)[/tex] are D and E.
