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Advanced Math / Nonlinear functions Difficulty: Hard

The functions $g$ and $h$ are defined by the given equations, where $x greater than or equals 0$ . Which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where $x greater than or equals 0$ ?

$g (x) = 18 (1.16) {(1.4)}^{x + 2}$
$h (x) = 18 {(1.4)}^{x + 4}$

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Explanation

Choice D is correct. A function defined by an equation in the form $f (x) = a {(b)}^{x + h}$ , where $a$ , $b$ , and $h$ are positive constants and $x greater than or equals 0$ , has a minimum value of $f left parenthesis 0 right parenthesis$ . It's given that function $g$ is defined by $g (x) = 18 (1.16) {(1.4)}^{x + 2}$ , which is equivalent to $g (x) = 20.88 {(1.4)}^{x + 2}$ . Substituting $0$ for $x$ in this equation yields $g (0) = 20.88 {(1.4)}^{0 + 2}$ , or $g left parenthesis 0 right parenthesis equals 40.9248$ . Therefore, the minimum value of $g (x)$ is $40.9248$ , so $g (x) = 18 (1.16) {(1.4)}^{x + 2}$ doesn't display its minimum value as a constant or coefficient. It's also given that function $h$ is defined by $h (x) = 18 {(1.4)}^{x + 4}$ . Substituting $0$ for $x$ in this equation yields $h (0) = 18 {(1.4)}^{0 + 4}$ , or $h left parenthesis 0 right parenthesis equals 69.1488$ . Therefore, the minimum value of $h (x)$ is $69.1488$ , so $h (x) = 18 {(1.4)}^{x + 4}$ doesn't display its minimum value as a constant or coefficient. Therefore, neither I nor II displays, as a constant or coefficient, the minimum value of the function it defines, where $x greater than or equals 0$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.