Question

suppose that f is given for x in the interval 0,12 by

x =	0	2	4	6	8	10	12
f(x)=	17	15	13	10	9	10	12

a. estimate f(2) using the values of f in the table.
f(2)≈
b. for what values of x does f(x) appear to be positive?
(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)
c. for what values of x does f(x) appear to be negative?
(give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10).)
submit answer next item

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. $(0,8)$