Problem

Which statements describe the function $f(x)=2(x-4)^{4}$ ?
A. It has 4 zeros and at most 3 relative maximums or minimums.
B. It has 3 zeros and at most 4 relative maximums or minimums.
C. It is a translation of the parent function 4 units to the right.
D. Both ends of the graph of the function go up.
E. It is a translation of the parent function 4 units to the left.
F. The left end of the graph of the function goes up, and the right end goes down.

Answer

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Answer

Final Answer: \(\boxed{\text{C, D}}\)

Steps

Step 1 :The function \(f(x)=2(x-4)^{4}\) is a polynomial function of degree 4. It is a transformation of the parent function \(f(x)=x^{4}\), where the graph is shifted 4 units to the right and vertically stretched by a factor of 2.

Step 2 :The function has only one zero at x=4, because \((x-4)^{4}\) equals zero only when x=4.

Step 3 :The function has no relative maximums or minimums, because it is always increasing for x>4 and always decreasing for x<4.

Step 4 :Both ends of the graph of the function go up, because as x approaches positive or negative infinity, the value of the function also approaches positive infinity.

Step 5 :Therefore, the correct statements are C and D.

Step 6 :Final Answer: \(\boxed{\text{C, D}}\)

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