Question

Determine whether the function $f(x)=-x^{3}+5 x^{4}$ is even, odd or neither.
Answer
odd
neither
even

Step 1 ：A function is said to be even if the following condition is met: \(f(x) = f(-x)\) for all x in the function's domain. A function is said to be odd if the following condition is met: \(-f(x) = f(-x)\) for all x in the function's domain. If a function is neither even nor odd, then it does not meet either of these conditions.

Step 2 ：In this case, we need to check whether the function \(f(x) = -x^3 + 5x^4\) is even, odd, or neither. We can do this by substituting -x into the function and simplifying the result. If the simplified result is equal to the original function, then the function is even. If the simplified result is equal to the negative of the original function, then the function is odd. If the simplified result is neither of these, then the function is neither even nor odd.

Step 3 ：Let's substitute -x into the function and simplify the result.

Step 4 ：The result of substituting -x into the function is not equal to the original function, nor is it equal to the negative of the original function. Therefore, the function is neither even nor odd.

Step 5 ：Final Answer: The function \(f(x)=-x^{3}+5 x^{4}\) is \(\boxed{\text{neither even nor odd}}\).