Final answer:
To show that there is a root of the equation 7x³ - 9x² + 3x - 2 = 0 between 1 and 2, we can use the Intermediate Value Theorem.
Step-by-step explanation:
To show that there is a root of the equation 7x³ - 9x² + 3x - 2 = 0 between 1 and 2, we can use the Intermediate Value Theorem. The theorem states that if a function is continuous on a closed interval [a, b] and takes on values f(a) and f(b) where f(a) < 0 and f(b) > 0 (or vice versa), then there exists at least one c in the interval (a, b) where f(c) = 0.
Let's evaluate the function at both ends of the interval:
f(1) = 7(1)³ - 9(1)² + 3(1) - 2 = -1
f(2) = 7(2)³ - 9(2)² + 3(2) - 2 = 26
Since f(1) < 0 and f(2) > 0, by the Intermediate Value Theorem, there must be a root between 1 and 2. Therefore, the equation has a solution in that interval.