Questions: Determine End Behavior For the function f(x) = 1/2 x^3(x+4)(x-1), the leading term is positive and the degree is odd. Use this information to determine the end behavior of the function. Provide your answer below: as x → ∞, f(x) → as x → -∞, f(x) →

Solution Steps

To determine the end behavior of the function \( f(x) = \frac{1}{2} x^{3}(x+4)(x-1) \), we need to consider the leading term and the degree of the polynomial. The leading term is positive and the degree is odd. For polynomials with a positive leading coefficient and an odd degree, as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \) and as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \).

The function given is \( f(x) = \frac{1}{2} x^{3}(x+4)(x-1) \). The leading term of this polynomial is determined by the highest degree term, which is \( \frac{1}{2} x^{3} \). The degree of the polynomial is \( 3 \), which is odd, and the leading coefficient \( \frac{1}{2} \) is positive.

For polynomials, the end behavior can be analyzed based on the degree and the sign of the leading coefficient. Since the leading coefficient is positive and the degree is odd, we can conclude the following:

• As \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \).
• As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \).

Final Answer