Is the following function Odd or Even? $$ f(x)=-5 x^{4}-6 x^{3}+2 x $$

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To determine whether the function $$ f(x) = -5x^{4} - 6x^{3} + 2x $$ is **even**, **odd**, or **neither**, we can analyze its symmetry properties.

### Definitions:
- **Even Function:** A function $$ f(x) $$ is even if $$ f(-x) = f(x) $$ for all $$ x $$.
- **Odd Function:** A function $$ f(x) $$ is odd if $$ f(-x) = -f(x) $$ for all $$ x $$.

### Step-by-Step Analysis:

1. **Compute $$ f(-x) $$:**
   $$
   f(-x) = -5(-x)^4 - 6(-x)^3 + 2(-x)
   $$
   
   Simplifying each term:
   $$
   (-x)^4 = x^4 \quad \text{(since even exponent)}
   $$
   $$
   (-x)^3 = -x^3 \quad \text{(since odd exponent)}
   $$
   
   Substituting back:
   $$
   f(-x) = -5x^4 - 6(-x^3) + 2(-x) = -5x^4 + 6x^3 - 2x
   $$

2. **Compare $$ f(-x) $$ with $$ f(x) $$ and $$ -f(x) $$:**
   - **Even Check:** 
     $$
     f(-x) \stackrel{?}{=} f(x) \quad \Rightarrow \quad -5x^4 + 6x^3 - 2x \stackrel{?}{=} -5x^4 - 6x^3 + 2x
     $$
     This is **not** true for all $$ x $$.

- **Odd Check:** 
     $$
     f(-x) \stackrel{?}{=} -f(x) \quad \Rightarrow \quad -5x^4 + 6x^3 - 2x \stackrel{?}{=} 5x^4 + 6x^3 - 2x
     $$
     This is also **not** true for all $$ x $$.

3. **Conclusion:**
   The function $$ f(x) $$ contains both even ($$ x^4 $$) and odd ($$ x^3 $$ and $$ x $$) degree terms. This mixture means the function does not satisfy the symmetry conditions for being either even or odd.

### **Answer:**
Neither. The function contains both even and odd terms and does not exhibit even or odd symmetry.
The function $$ f(x) = -5x^{4} - 6x^{3} + 2x $$ is **neither even nor odd** because it contains both even and odd degree terms, which prevents it from satisfying the symmetry conditions required for even or odd functions.

Is the following function Odd or Even?

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