suppose that f is given for x in the interval 0,12 by\n|x = |0|2|4|6|8|10|12|\n|f(x)=|17|15|13|10|9|10|12|\na…

suppose that f is given for x in the interval 0,12 by\n|x = |0|2|4|6|8|10|12|\n|f(x)=|17|15|13|10|9|10|12|\na. estimate f(2) using the values of f in the table.\nf(2)≈\nb. for what values of x does f(x) appear to be positive?\n(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)\nc. for what values of x does f(x) appear to be negative?\n(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)\nsubmit answer next item

suppose that f is given for x in the interval 0,12 by\n|x = |0|2|4|6|8|10|12|\n|f(x)=|17|15|13|10|9|10|12|\na. estimate f(2) using the values of f in the table.\nf(2)≈\nb. for what values of x does f(x) appear to be positive?\n(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)\nc. for what values of x does f(x) appear to be negative?\n(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)\nsubmit answer next item

Answer

Explanation:

Step1: Estimate $f'(2)$ using difference quotient

The forward - difference quotient formula is $f'(a)\approx\frac{f(a + h)-f(a)}{h}$. Here $a = 2$ and $h=2$, so $f'(2)\approx\frac{f(4)-f(2)}{4 - 2}$. We know that $f(2)=15$ and $f(4)=13$. Then $f'(2)\approx\frac{13 - 15}{2}=\frac{-2}{2}=-1$.

Step2: Determine where $f'(x)>0$

A function $y = f(x)$ has a positive derivative $f'(x)>0$ when the function is increasing. Looking at the table, $f(x)$ is increasing on the interval $(8,12)$ since $f(8) = 9$, $f(10)=10$ and $f(12)=12$.

Step3: Determine where $f'(x)<0$

A function $y = f(x)$ has a negative derivative $f'(x)<0$ when the function is decreasing. From the table, $f(x)$ is decreasing on the intervals $(0,8)$ since $f(0)=17$, $f(2)=15$, $f(4)=13$, $f(6)=10$ and $f(8)=9$.

Answer:

A. $-1$ B. $(8,12)$ C. $(0,8)$