determining the interval where the function is increasing\nusing only the values given in the table for the…

determining the interval where the function is increasing\nusing only the values given in the table for the function $f(x)=-x^{3}+4x + 3$, what is the largest interval of $x$-values where the function is increasing?\n|$x$|$f(x)$|\n|-3|18|\n|-2|3|\n|-1|0|\n|0|3|\n|1|6|\n|2|3|

determining the interval where the function is increasing\nusing only the values given in the table for the function $f(x)=-x^{3}+4x + 3$, what is the largest interval of $x$-values where the function is increasing?\n|$x$|$f(x)$|\n|-3|18|\n|-2|3|\n|-1|0|\n|0|3|\n|1|6|\n|2|3|

Answer

Explanation:

Step1: Recall increasing - function definition

A function $y = f(x)$ is increasing if $f(x_1)<f(x_2)$ when $x_1 < x_2$.

Step2: Analyze the table values

When $x=-3$, $f(-3)=18$; when $x = - 2$, $f(-2)=3$ (decreasing). When $x=-2$, $f(-2)=3$; when $x=-1$, $f(-1)=0$ (decreasing). When $x=-1$, $f(-1)=0$; when $x = 0$, $f(0)=3$ (increasing). When $x = 0$, $f(0)=3$; when $x=1$, $f(1)=6$ (increasing). When $x = 1$, $f(1)=6$; when $x=2$, $f(2)=3$ (decreasing).

Step3: Determine the increasing interval

The function is increasing for $x$ values from $- 1$ to $1$.

Answer:

$(-1,1)$