Question

the graphs of $f(x)=100cdot1.5^{x}$ and $g(x)=100x + 100$ is shown. which of the following statements is true?
statement #1: while the growth rate of $f(x)$ is initially greater than the growth rate of $g(x)$, the growth rate of $g(x)$ keeps increasing and eventually surpasses the growth rate of $f(x)$.
statement #2: the growth rate of $g(x)$ is greater than the growth rate of $f(x)$ between approximately $x = 0$ and $x = 4$
statement #3: while the growth rate of $g(x)$ is initially greater than the growth rate of $f(x)$, the growth rate of $f(x)$ keeps increasing and by $x = 2.5$, surpasses the growth rate of $g(x)$

Explanation:

Step1: Analyze the growth rate of \(f(x) = 100\times1.5^{x}\)

The function \(f(x)=100\times1.5^{x}\) is an exponential function. The growth rate of an exponential function \(y = a\times b^{x}\) (\(a>0,b > 1\)) is given by its derivative \(y^\prime=a\times b^{x}\times\ln(b)\). For \(f(x)=100\times1.5^{x}\), \(f^\prime(x)=100\times1.5^{x}\times\ln(1.5)\). As \(x\) increases, \(f^\prime(x)\) increases because \(1.5^{x}\) increases as \(x\) increases.

Step2: Analyze the growth rate of \(g(x)=100x + 100\)

The function \(g(x)=100x + 100\) is a linear function. The growth rate of a linear function \(y=mx + c\) is given by its slope. For \(g(x)=100x + 100\), the derivative \(g^\prime(x)=100\) (a constant).

Step3: Compare the growth rates

• Initially (for small \(x\) values, say \(x = 0\)), \(f^\prime(0)=100\times1\times\ln(1.5)\approx100\times0.405 = 40.5\) and \(g^\prime(x)=100\). So, the growth rate of \(g(x)\) is greater than the growth rate of \(f(x)\) for small \(x\) values.
• As \(x\) increases, let's find when \(f^\prime(x)>g^\prime(x)\). We set \(100\times1.5^{x}\times\ln(1.5)>100\). Divide both sides by \(100\) (since \(100>0\)), we get \(1.5^{x}\times\ln(1.5)>1\). \(1.5^{x}>\frac{1}{\ln(1.5)}\approx2.47\). Take the natural - logarithm of both sides: \(x\ln(1.5)>\ln(2.47)\), \(x>\frac{\ln(2.47)}{\ln(1.5)}\approx2.5\).

Answer:

Statement #2: The growth rate of \(g(x)\) is greater than the growth rate of \(f(x)\) between approximately \(x = 0\) and \(x = 4\) is true.