Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.

Question

f(x) = 6x³+9x²-3x+3; [-2, -1]

Find the value of f(-2).

f(-2) =

(Simplify your answer.)

Answer 1

The intermediate value theorem state that, if f is continuous from a to b and f(a)f(b) < 0, then there exists at least one x between a and b such that f(x) = 0.

As f is a polynomial, f is continuous everywhere, including between a = -2 and b = -1

f(a) = f(-2) = 6 * (-2)³ + 9 * (-2)² - 3 * (-2) + 3 = -3

f(b) = f(-1)= 6 * (-1)³ + 9 * (-1)² - 3 * (-1) + 3 = 9

f(a)f(b) = (-3)(9) = -27 < 0

Thus, there is a 0 between -2 and-1