Marshall's mistake arises from comparing the growth rates of the two functions without fully understanding their behaviors, especially as \( x \) approaches larger values.
To assess which function grows faster, we need to examine the growth rates analytically:
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Exponential Function: \( f(x) = 6 \times 1.5^x \)
- This function is exponential because it has a base of 1.5 raised to the power of \( x \). Exponential functions generally grow faster than polynomial functions as \( x \) becomes large.
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Linear Function: \( g(x) = 5x + 10 \)
- This is a linear function, which grows at a constant rate. As \( x \) increases, \( g(x) \) increases linearly.
To compare these functions more rigorously, we should evaluate their limits as \( x \) approaches infinity:
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For \( f(x) \): \[ \lim_{x \to \infty} f(x) \text{ grows without bound since it is exponential.} \]
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For \( g(x) \): \[ \lim_{x \to \infty} g(x) \text{ also grows without bound but at a linear rate.} \]
To see the growth rates, we can look at the derivatives of each function:
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The derivative of \( f(x) = 6 \times 1.5^x \) is: \[ f'(x) = 6 \times 1.5^x \ln(1.5) \] This shows that \( f'(x) > 0 \) and is increasing for all \( x \).
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The derivative of \( g(x) = 5x + 10 \) is: \[ g'(x) = 5 \] This is constant.
For sufficiently large \( x \), \( f'(x) \) (which represents the rate of growth of \( f(x) \)) will exceed \( g'(x) \). So, as \( x \) becomes large enough, \( f(x) \) will grow faster than \( g(x) \).
The crucial point in Marshall's analysis is the misunderstanding that at all \( x \), \( g(x) \) is greater than \( f(x) \). While it may be true for small values of \( x \) (where the linear term could dominate due to its initial values), eventually, as \( x \) increases, the exponential function \( f(x) = 6 \times 1.5^x \) will outpace the linear function \( g(x) = 5x + 10 \).
Therefore, Marshall's conclusion overlooks the asymptotic behavior of the functions, where the exponential function eventually grows faster than the linear one for large \( x \).
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