select the correct answer. consider functions f and g. f(x)=x³ + 5x² - x which statement is true about these…

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 the same rate that 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 a faster rate than function g is decreasing. d. over the interval -2, 2, function f is decreasing at a faster rate than function g is increasing.

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 the same rate that 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 a faster rate than function g is decreasing. d. over the interval -2, 2, function f is decreasing at a faster rate than function g is increasing.

Answer

Explanation:

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

$f(x)=x^{3}+5x^{2}-x$, using the power - rule $(x^n)' = nx^{n - 1}$, we have $f'(x)=3x^{2}+10x - 1$.

Step2: Calculate the change in $f(x)$ over $[-2,2]$

$f(-2)=(-2)^{3}+5\times(-2)^{2}-(-2)=-8 + 20+2 = 14$. $f(2)=2^{3}+5\times2^{2}-2=8 + 20 - 2=26$. The change in $f(x)$ is $\Delta f=f(2)-f(-2)=26 - 14 = 12$.

Step3: Calculate the change in $g(x)$ over $[-2,2]$

$g(-2)=-4$, $g(2)=-16$. The change in $g(x)$ is $\Delta g=g(2)-g(-2)=-16-(-4)=-12$. The magnitude of the change in $f(x)$ and $g(x)$ over the interval $[-2,2]$ is the same.

Answer:

A. Over the interval $[-2,2]$, function $f$ is increasing at the same rate that function $g$ is decreasing.