Show that cos x = x^3+x^2+4x has exactly one root in [0, 3.14.../2]
Let .
Use the Intermediate Value Theorem to show that f(x) has at least one root in the interval .
Now assume f(x) has two roots, a and b say, in the interval . 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 ).