Show that hes exactly one root in .
On the interval $\displaystyle (0, 0.5\pi)$, the function $\displaystyle f(x) = x^3 + x^2 + 4x - cos x$ is negative at $\displaystyle x = 0$ but positive at $\displaystyle x = 0.5\pi$. If you can show that the derivative of f(x) is positive for $\displaystyle x \in (0, 0.5\pi)$, then you know that a root exists by the Intermediate Value Theorem, and that it is the only root by the Extreme Value Theorem.