To analyze the end behavior of the function [tex]\( f(x) = 5x^3 - 3x + 332 \)[/tex], let's break down the components and properties of this polynomial.
1. Identify the leading term:
The function [tex]\( f(x) \)[/tex] is a polynomial, and the term with the highest power of [tex]\( x \)[/tex] dictates the end behavior of the function. Here, the leading term is [tex]\( 5x^3 \)[/tex].
2. Consider the leading coefficient:
In the term [tex]\( 5x^3 \)[/tex], the leading coefficient is 5, which is positive.
3. Examine the degree of the polynomial:
The degree of the polynomial is the highest power of [tex]\( x \)[/tex], which in this case is 3. Since 3 is odd, the end behavior will differ at [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex].
Based on these components:
- When [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] approaches positive infinity), the term [tex]\( 5x^3 \)[/tex] grows rapidly in the positive direction because the coefficient 5 is positive. Therefore, [tex]\( f(x) \)[/tex] will also tend to [tex]\( \infty \)[/tex].
- When [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] approaches negative infinity), the term [tex]\( 5x^3 \)[/tex] grows rapidly in the negative direction (since [tex]\( x^3 \)[/tex] becomes more negative and is multiplied by 5). Therefore, [tex]\( f(x) \)[/tex] will tend to [tex]\( -\infty \)[/tex].
Hence, for an odd-degree polynomial with a positive leading coefficient, the left end (as [tex]\( x \to -\infty \)[/tex]) goes down (towards negative infinity) and the right end (as [tex]\( x \to \infty \)[/tex]) goes up (towards positive infinity).
This perfectly matches option:
D. The leading coefficient is positive so the left end goes down.
Therefore, the correct choice is [tex]\(\boxed{D}\)[/tex].
