Give an appropriate answer.Show that the function f(x) = x3 + + 2 has exactly one zero on the interval (-?, 0).

Question

What will be an ideal response?

Answer 1

The function f(x) is continuous on the open interval (-?, 0). Also, f(x) approaches -? as x approaches -?, and f(x) approaches ? as x approaches 0 from the left. Since f(x) is continuous and changes sign along the interval, it must have at least one root on the interval.

The first derivative of f(x) is f'(x) = 3x² - , which is everywhere positive on (-?, 0). Thus, f(x) has a single root on (-?, 0).