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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 $x greater than or equals 0$ ?

$f (x) = 33 {(0.4)}^{x + 3}$
$g (x) = 33 (0.16) {(0.4)}^{x - 2}$

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Explanation

Choice B is correct. Functions $f$ and $g$ are both exponential functions with a base of $0.40$ . Since $0.40$ is less than $1$ , functions $f$ and $g$ are both decreasing exponential functions. This means that $f (x)$ and $g (x)$ decrease as $x$ increases. Since $f (x)$ and $g (x)$ decrease as $x$ increases, the maximum value of each function occurs at the least value of $x$ for which the function is defined. It's given that functions $f$ and $g$ are defined for $x greater than or equals 0$ . Therefore, the maximum value of each function occurs at $x equals 0$ . Substituting $0$ for $x$ in the equation defining $f$ yields $f (0) = 33 {(0.4)}^{0 + 3}$ , which is equivalent to $f left parenthesis 0 right parenthesis equals 33 left parenthesis 0.4 right parenthesis cubed$ , or $f left parenthesis 0 right parenthesis equals 2.112$ . Therefore, the maximum value of $f$ is $2.112$ . Since the equation $f (x) = 33 {(0.4)}^{x + 3}$ doesn't display the value $2.112$ , the equation defining $f$ doesn't display the maximum value of $f$ . Substituting $0$ for $x$ in the equation defining $g$ yields $g (0) = 33 (0.16) {(0.4)}^{0 - 2}$ , which can be rewritten as $g (0) = 33 (0.16) (\frac{1}{{0.4}^{2}})$ , or $g (0) = 33 (0.16) (\frac{1}{0.16})$ , which is equivalent to $g left parenthesis 0 right parenthesis equals 33$ . Therefore, the maximum value of $g$ is $33$ . Since the equation $g (x) = 33 (0.16) {(0.4)}^{x - 2}$ displays the value $33$ , the equation defining $g$ displays the maximum value of $g$ . Thus, only equation II displays, as a constant or coefficient, the maximum value of the function it defines.

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.