How do we find whether the function #f(x)=cosx-x^2# has a root between #x=pi/4# and #x=pi/3# or not?

When we have two points connected by a continuous curve, one point below the line and the other above the line, then according to Intermediate Value Theorem, there will be at least one point where the curve crosses the line.

and hence,

To determine if the function $f(x) = \cos(x) - x^2$ has a root between $x = \frac{\pi}{4}$ and $x = \frac{\pi}{3}$, we can evaluate the function at these two values and observe if the signs of the function values differ. If the sign changes between these two points, then there is at least one root within the interval.

1. Evaluate $f\left(\frac{\pi}{4}\right)$ and $f\left(\frac{\pi}{3}\right)$.
2. Check if the signs of $f\left(\frac{\pi}{4}\right)$ and $f\left(\frac{\pi}{3}\right)$ are different.

If the signs are different, then there exists at least one root between $x = \frac{\pi}{4}$ and $x = \frac{\pi}{3}$. If the signs are the same, then there may or may not be a root in that interval, and further investigation is required.