AP Calculus AB Limits and Continuity Question

Question

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Can someone please help me find the answer and explain it? Thank you.

Answer 1

It is given that the polynomial f(x) = x³ - 2x + 5 has only one real root.

Calculate values:

f(-4) = -51, f(-2) = 1

( f(-1) = 6, f(1) = 4, f(3) = 26, but actually we do not need these values)

Because f(-4) < 0, f(-2) > 0, and polynomials are continuous functions, there exists a number c in (-4, -2) such that f(c) = 0 (The Intermediate Value Theorem).

Because this polynomial has only one real root, so the answer is (A) between -4 and -2.