Use the Intermediate Value Theorem (IVT) to show that f (x) = −x5−4x2+ 5 has a root on the interval (0, 2).

Answer 1

Because f(x) is a polynomial function, it is continuous over the real numbers, which means the IVT applies to f(x) on the interval [0 , 2].

f(0) = 5 and f(2) = - 43 so the IVT guarantees that f(x) will attain every value between - 43 and 5 at least once for an x-value with 0 < x < 2.

Finally, since - 43 < 0 < 5, f(x) has a root (i.e. an x₀ such that f(x₀) = 0) on the interval (0 , 2).

Answer 2

f(0) = ?

f(2) = ?

What does the Intermediate Value Theorem state?