Question

select the correct answer.
consider the functions ( f ) and ( g ) in the tables below.

which of the following statements is true?
a. for every value of ( x ), the rate of change of ( g ) exceeds the rate of change of ( f ).
b. at approximately ( x = 4.39 ), the rate of change of ( f ) is equal to the rate of change of ( g ).
c. as ( x ) increases, the rate of change of ( g ) exceeds the rate of change of ( f ).
d. as ( x ) increases, the rate of change of ( f ) exceeds the rate of change of ( g ).

Explanation:

Step1: Find the derivative of \(f(x)\)

The derivative of \(f(x)=90x^{2}+180x + 92\) using the power rule \((x^{n})^\prime=nx^{n - 1}\):
\(f^\prime(x)=\frac{d}{dx}(90x^{2}+180x + 92)=180x+180\)

Step2: Find the derivative of \(g(x)\)

The derivative of \(g(x)=6^{x}\) using the formula \((a^{x})^\prime=a^{x}\ln a\) (\(a = 6\)):
\(g^\prime(x)=\frac{d}{dx}(6^{x})=6^{x}\ln6\)

Step3: Analyze the behavior of the derivatives as \(x\) increases

• For \(x = 0\):

\(f^\prime(0)=180\times0 + 180=180\), \(g^\prime(0)=6^{0}\ln6=\ln6\approx1.79\)

• As \(x\) increases:

The function \(y = f^\prime(x)=180x + 180\) is a linear function with a slope of \(180\).
The function \(y = g^\prime(x)=6^{x}\ln6\) is an exponential function. The exponential function \(y = a^{x}\) (\(a>1\)) grows without bound and will eventually over - take any linear function.

Answer:

C. As \(x\) increases, the rate of change of \(g\) exceeds the rate of change of \(f\).