Show that cos x = x^3+x^2+4x has exactly one root in [0, 3.14.../2]
Let $\displaystyle f(x) = x^3 + x^2 + 4x - \cos x$.
Use the Intermediate Value Theorem to show that f(x) has at least one root in the interval $\displaystyle \left[ 0, \, \frac{\pi}{2}\right]$.
Now assume f(x) has two roots, a and b say, in the interval $\displaystyle \left[ 0, \, \frac{\pi}{2}\right]$. Then f(a) = f(b) = 0. Now use Rolles Theorem on the closed interval [a, b] to show that this leads to an impossibility ..... (Big Hint: Consider the turning point of $\displaystyle y = 3x^2 + 2x + 4$).