$(7$ points$)$ Circle exactly one phrase per box below to complete the statement of the Intermediate Value Theorem.

$f(a),f(b)$

$(f(a),f(b))$

$a,b$

$(a,b)$

$f(a),f(b)$

$a,b$

$(f(a),f(b))$

$(a,b)$

the max and min of $f(x)$

a and $b$

$f(a)$ and $f(b)$

$-K$ and $K$

$f(K)=c$

$f(c)=K$

$f(K)=a$

$f(c)=a$

$(4$ points$)$ Let $f(x)=x^3-3x^2+1.$

$($a$)$ Explain why the the Intermediate Value Theorem can be applied to $f$ on the closed interval $1,3.$

$($b$)$ Then use the Intermediate Value Theorem to show $f$ has a root in the open interval $(1,3).$