To prove that the function f(x) = x^3 + 3/(x^2) + 2 has exactly one zero on the interval (-∞, 0), you can use the Intermediate Value Theorem. Here's how:
1. Start by rewriting the equation f(x) = 0:
x^3 + 3/(x^2) + 2 = 0
2. Notice that the function f(x) is continuous on the interval (-∞, 0) since it is a polynomial and rational function.
3. Evaluate the values of f(x) at the endpoints of the interval:
f(0) = 0^3 + 3/(0^2) + 2 = 2
f(-1) = (-1)^3 + 3/((-1)^2) + 2 = -2 + 3 + 2 = 3
4. Since f(0) = 2 and f(-1) = 3, the function passes through the values 2 and 3 on the interval (-∞, 0).
5. According to the Intermediate Value Theorem, if a function is continuous on an interval and takes on two distinct values at the endpoints, then it must take on every value in between.
6. Since f(x) goes from positive (2) to negative (3) as x goes from 0 to -1, it must pass through zero at some point in between.
Therefore, the function f(x) = x^3 + 3/(x^2) + 2 has exactly one zero on the interval (-∞, 0).
You were on the right track by setting f(x) = 0, but solving the resulting equation analytically can be difficult. Instead, using the Intermediate Value Theorem is a more appropriate method for proving the existence of a zero in this case.