Chapter 2 - Limits - 2.8 Intermediate Value Theorem - Exercises - Page 86: 3

See the proof below.

Since the function $ g(t)=t^2\tan t $ is continuous on $[0,\frac{\pi}{4}]$ and $ g(0)=0\neq g(\frac{\pi}{4})=\frac{\pi^2}{16}$ and $\frac{1}{2}$ is between $ g(0)$ and $ g(\frac{\pi}{4})$, then by the Intermediate Value Theorem the function $ g(t)$ takes on the value $\frac{1}{2}$ for some $ t\in (0,\frac{\pi}{4})$.