Question

If $f\left(x\right)$ is continuous on $[a,b]$ and $f\left(a\right)\ne f\left(b\right)$ , then for
any value $c$ belongs $\left(f\right(a),f(b\left)\right)$ , there is at least one number
$x_o$ in $(a,b)$ for which $f\left(x_o\right)=c$

• A. $f\left(x_o\right)=b$
• B. $f\left(x_o\right)=c$
• C. $f\left(x_o\right)=f\left(c\right)$
• D. $f\left(x_o\right)=0$

Answer 1

The correct option is B $f\left(x_o\right)=c$

Given that $f\left(x\right)$ is continuous

$\Rightarrow f\left(x\right)$ is a continuous (no holes)

curve from $f\left(a\right)$ to $f\left(b\right)$

There exists some value of x satisfying each point between $f\left(a\right)$ and $f\left(b\right)$

$\Rightarrow f\left(x_0\right)=c$

$\Rightarrow {lim}_{x\to x_0}f\left(x\right)=c$