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.

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.

Let T be the statement: For all real numbers x and y with x < y, there exists an integer n such that x ≤ n ≤ y.Write the negation of T.

let f:X -> (0,inf) be a continuous function on a compact space X. show that there exits an ε >0 such that f(x)>=ε for all x element of X

) Let f, g be differentiable functions on an interval I. Suppose that a, b ∈ Iwith f (b) ≥ g(b) and f ′(x) ≤ g′(x) on [a, b]. Show that f (x) ≥ g(x) on[a, b].

Is there a number a such that the following limit exists? (If an answer does not exist, enter DNE.) lim x→−2 4x2 + ax + a + 12x2 + x − 2 Find the value a.