Question

the polynomial function f is given by f(x)=ax^4 + bx^3+cx^2+dx + k, where a≠0 and b, c, d, and k are constants. which of the following statements about f is true? a f has both a global maximum and a global minimum. b f has either a global maximum or a global minimum, but not both. c f has neither a global maximum nor a global minimum. d the nature of a global maximum or a global minimum for f cannot be determined without more information about b, c, d and k

Explanation:

Step1: Analyze the degree of the polynomial

The function $f(x)=ax^{4}+bx^{3}+cx^{2}+dx + k$ is a polynomial of degree $n = 4$ (even - degree polynomial) with $a
eq0$.

Step2: Consider the end - behavior

For an even - degree polynomial $y = f(x)$ with leading coefficient $a$, as $x\to\pm\infty$, if $a>0$, then $y\to+\infty$; if $a < 0$, then $y\to-\infty$.
If $a>0$, the function has a global minimum and no global maximum. If $a < 0$, the function has a global maximum and no global minimum. So the function $f(x)$ has either a global maximum or a global minimum, but not both.

Answer:

B. $f$ has either a global maximum or a global minimum, but not both.