IMathAS-libretexts

Suppose

\displaystyle {f}'{\left({x}\right)}

is continuous over an interval

\displaystyle {a}\le{x}\le{b}

, and

\displaystyle {a}<{c}<{d}<{b}

.

If

\displaystyle {f}'{\left({c}\right)}={10}

, then (check all that apply)

$\displaystyle {f{{\left({c}\right)}}}$ exists
$\displaystyle {f{{\left({x}\right)}}}$ is decreasing through $\displaystyle {x}={c}$
$\displaystyle {f{{\left({x}\right)}}}$ is increasing through $\displaystyle {x}={c}$
$\displaystyle {f}$ does not have a maximum at $\displaystyle {x}={c}$
$\displaystyle {f{{\left({c}\right)}}}={10}$
$\displaystyle {f}$ does not have a minimum at $\displaystyle {x}={c}$

\displaystyle {f}'{\left({d}\right)}=-{6}

, then (check all that apply)

$\displaystyle {f}$ does not have a maximum at $\displaystyle {x}={d}$
$\displaystyle {f}$ does not have a minimum at $\displaystyle {x}={d}$
$\displaystyle {f{{\left({d}\right)}}}$ exists
$\displaystyle {f{{\left({d}\right)}}}=-{6}$
$\displaystyle {f{{\left({x}\right)}}}$ is increasing through $\displaystyle {x}={d}$
$\displaystyle {f{{\left({x}\right)}}}$ is decreasing through $\displaystyle {x}={d}$

Based on this, we know that for some

\displaystyle {x}

between

\displaystyle {x}={c}

and

\displaystyle {x}={d}