the function f is given by f(x)= -2x^7 + 5x^4+6x^2 - 3. which of the following correctly describes the end…

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function (f(x)=-2x^{7}+5x^{4}+6x^{2}-3) is (-2x^{7}) since the term with the highest power of (x) is (-2x^{7}) and its degree (n = 7) (odd) and coefficient (a=-2) (negative).

Step2: Analyze the end - behavior as (x\to+\infty)

For a polynomial function (y = a_nx^n+\cdots+a_0) with (n) odd and (a_n<0), as (x\to+\infty), we consider the behavior of the leading - term. When (x\to+\infty), (y=-2x^{7}\to-\infty) because a positive number (x) raised to an odd power (x^{7}) is positive, and multiplying by (- 2) makes it negative.

Step3: Analyze the end - behavior as (x\to-\infty)

When (x\to-\infty), (x^{7}\to-\infty) (since (n = 7) is odd). Then (y=-2x^{7}\to+\infty) because a negative number (x^{7}) multiplied by (-2) (a negative coefficient) gives a positive result.

Answer:

(\lim_{x\to+\infty}f(x)=-\infty)