Question

question 14
the function f(x) is continuous on the interval 0,8. the table below gives some of its values. what is the minimum number of zeros that f(x) is guaranteed to have by the intermediate value theorem?

x	f(x)
1	9
2	-10
3	-5
4	-6
5	-6
6	7
7	8
8	7

options:
4 zeros
2 zeros
0 zeros
3 zeros
1 zero
5 zeros

Explanation:

Step1: Check sign changes (0 to 1)

$f(0)=-1$, $f(1)=9$: sign changes from negative to positive, so there is at least 1 zero in $(0,1)$.

Step2: Check sign changes (1 to 2)

$f(1)=9$, $f(2)=-10$: sign changes from positive to negative, so there is at least 1 zero in $(1,2)$.

Step3: Check sign changes (2 to 3)

$f(2)=-10$, $f(3)=-5$: no sign change, no guaranteed zero.

Step4: Check sign changes (3 to 6)

$f(3)=-5$, $f(4)=-6$, $f(5)=-6$, $f(6)=7$: sign changes from negative to positive at $(5,6)$, so there is at least 1 zero in $(5,6)$.

Step5: Check sign changes (6 to 8)

$f(6)=7$, $f(7)=8$, $f(8)=7$: no sign change, no guaranteed zero.

Step6: Count guaranteed zeros

Total guaranteed zeros: 1 + 1 + 1 = 3.

Answer:

3 zeros