Let f(x) = −x5 −2x4 + 12x3 +x−1. Then either show that there is an x such thatf(x) = 0, and find an interval where it exists; or, show there is no such x.

To solve this problem, we can use the Intermediate Value Theorem (IVT). The IVT states that if a function f is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.

Let's apply this theorem to the function f(x) = -x^5 - 2x^4 + 12x^3 + x - 1 and try to find an interval where f(x) = 0.

First, we need to find two values a and b such that f(a) < 0 and f(b) > 0 or f(a) > 0 and f(b) < 0.

Let's try a = -2 and b = 2.

f(-2) = -(-2)^5 - 2(-2)^4 + 12(-2)^3 + (-2) - 1 = -32 - 32 - 96 - 2 - 1 = -163 < 0

f(2) = -(2)^5 - 2(2)^4 + 12(2)^3 + 2 - 1 = -32 - 32 + 96 + 2 - 1 = 33 > 0

Since f(-2) < 0 and f(2) > 0, by the Intermediate Value Theorem, there exists at least one number c in the interval (-2, 2) such that f(c) = 0.

So, there is an x such that f(x) = 0, and it exists in the interval (-2, 2).