2 select the correct answer. consider functio...
2 select the correct answer. consider functions f and g. f(x)=x³ + 5x² - x which statement is true about these functions? a. over the interval -2, 2, function f is increasing at a faster rate than function g is decreasing. b. over the interval -2, 2, function f and function g are decreasing at the same rate. c. over the interval -2, 2, function f is increasing at the same rate that function g is decreasing. d. over the interval -2, 2, function f is decreasing at a faster rate than function g is increasing. x -2 -1 0 1 2 3 g(x) -4 8 6 2 -16 -84
Answer
# Explanation:
## Step1: Find the rate - of - change of function \(f(x)\) over \([-2,2]\)
The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). For \(f(x)=x^{3}+5x^{2}-x\), \(a=-2\) and \(b = 2\). First, find \(f(-2)\) and \(f(2)\):
\[
\begin{align*}
f(-2)&=(-2)^{3}+5\times(-2)^{2}-(-2)\\
&=-8 + 20+2\\
&=14
\end{align*}
\]
\[
\begin{align*}
f(2)&=2^{3}+5\times2^{2}-2\\
&=8 + 20-2\\
&=26
\end{align*}
\]
The average rate of change of \(f(x)\) over \([-2,2]\) is \(\frac{f(2)-f(-2)}{2-(-2)}=\frac{26 - 14}{4}=\frac{12}{4}=3\).
## Step2: Find the rate - of change of function \(g(x)\) over \([-2,2]\)
From the table, when \(x=-2\), \(g(-2)=-4\) and when \(x = 2\), \(g(2)=-16\). The average rate of change of \(g(x)\) over \([-2,2]\) is \(\frac{g(2)-g(-2)}{2-(-2)}=\frac{-16-(-4)}{4}=\frac{-16 + 4}{4}=\frac{-12}{4}=-3\). The magnitude of the rate of change of \(g(x)\) is \(3\).
# Answer:
C. Over the interval \([-2,2]\), function \(f\) is increasing at the same rate that function \(g\) is decreasing.