Answer
Assuming that the cooling function is both continuous on $[0,5]$ and differentiable on $(0,5)$,
the Mean Value Theorem applies,
and guarantees such a time $t_{0}$.
Work Step by Step
The Mean ValueTheorem
If $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ ,
then there exists a number $c$ in $(a, b)$ such that$ f^{\prime}(c)=\displaystyle \frac{f(b)-f(a)}{b-a}.$
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Let $f(t)$ be the temperature of the object.
$f(0)=1500^{o}$ and $f(5)=390^{o}$.
The average temperature over the interval $[0,5]$ is
$T_{avg}=\displaystyle \frac{f(5)-f(0)}{5-0}=\frac{390-1500}{5-0}=-222^{o}\mathrm{F}/\mathrm{h}.$
We assume that the cooling function is both continuous on $[0,5]$ and differentiable on $(0,5).$
By the Mean Value Theorem, a time $t_{0}$ exists such that
$0
