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Advanced Math / Nonlinear functions Difficulty: Hard

The functions $f$ and $g$ 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 maximum value of the function it defines, where $x greater than or equals 0$ ?

$f (x) = 18 {(1.25)}^{x} + 41$
$g left parenthesis x right parenthesis equals 9 left parenthesis 0.73 right parenthesis Superscript x$

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Explanation

Choice B is correct. For the function $f$ , since the base of the exponent, $1.25$ , is greater than $1$ , the value of $left parenthesis 1.25 right parenthesis Superscript x$ increases as $x$ increases. Therefore, the value of $18 left parenthesis 1.25 right parenthesis Superscript x$ and the value of $18 left parenthesis 1.25 right parenthesis Superscript x plus 41$ also increase as $x$ increases. Since $f$ is therefore an increasing function where $x greater than or equals 0$ , the function $f$ has no maximum value. For the function $g$ , since the base of the exponent, $0.73$ , is less than $1$ , the value of $left parenthesis 0.73 right parenthesis Superscript x$ decreases as $x$ increases. Therefore, the value of $9 left parenthesis 0.73 right parenthesis Superscript x$ also decreases as $x$ increases. It follows that the maximum value of $g (x)$ for $x greater than or equals 0$ occurs when $x equals 0$ . Substituting $0$ for $x$ in the function $g$ yields $g left parenthesis 0 right parenthesis equals 9 left parenthesis 0.73 right parenthesis Superscript 0$ , which is equivalent to $g left parenthesis 0 right parenthesis equals 9 left parenthesis 1 right parenthesis$ , or $g left parenthesis 0 right parenthesis equals 9$ . Therefore, the maximum value of $g (x)$ for $x greater than or equals 0$ is $9$ , which appears as a coefficient in equation II. So, of the two equations given, only II displays, as a constant or coefficient, the maximum 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 C is incorrect and may result from conceptual or calculation errors.

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