Locating roots by change of sign Quiz

1. $f(x)=x^2-5$. Between which integers does the positive root lie?

• A. $1$ and $2$
• B. $1$ and $3$
• C. $2$ and $3$
• D. $3$ and $4$

2. Why does $f(x)=\frac1x$ change sign between $-1$ and $1$ without having a root?

• A. It is not continuous at $x=0$
• B. It is a repeated root
• C. $f(0)=0$
• D. The interval is too wide for bisection

3. $f(x)=x^2-4x+4$ has $f(1)=1$ and $f(3)=1$. What is going on?

• A. No root exists
• B. Two distinct roots lie in $[1,3]$
• C. $f$ is discontinuous
• D. There is a repeated root at $x=2$

4. $f$ is continuous, $f(a)>0$ and $f(b)<0$. Which is true?

• A. There is exactly one root in $(a,b)$
• B. There is an odd number of roots in $(a,b)$
• C. There is no root in $(a,b)$
• D. There are two roots in $(a,b)$

5. Which is a complete argument that $f(x)=x^3+x-3$ has a root between 1 and 2?

• A. $f(1)=-1$ and $f(2)=7$
• B. $f(1)=-1$, $f(2)=7$, so there is a root
• C. $f(1)=-1<0$, $f(2)=7>0$, $f$ is continuous, so there is a root between 1 and 2
• D. The graph crosses the axis