Applying the IVT to show a specific temperature occurs

First, note that $T(x) = 60 - 5x + \frac{x^2}{2}$ is a polynomial, so it is continuous on every real interval, including the closed interval $[0,10]$. This satisfies the hypothesis of the IVT that $f$ must be continuous on $[a,b]$.

Next, evaluate $T$ at the endpoints and an interior point to find an interval where 50 is bracketed: $T(0) = 60 - 0 + 0 = 60$ and $T(5) = 60 - 5(5) + \frac{5^2}{2} = 60 - 25 + 12.5 = 47.5$.

Since $T$ is continuous on $[0,5]$, $T(0) = 60$, $T(5) = 47.5$, and $50$ lies strictly between $47.5$ and $60$, the IVT guarantees a value $c$ in $(0,5) \subset (0,10)$ with $T(c) = 50$. This shows the rod reaches exactly 50 degrees at some point along its length.

The solution applied the Intermediate Value Theorem on the interval $[0, 5]$. Since $T(0) = 60$ and $T(5) = 47.5$, the target value 50 lies strictly between them. This interval was chosen because it directly brackets the target value 50, allowing the IVT to guarantee the existence of $c$ in $(0, 5)$ with $T(c) = 50$.