Verify the given function has a zero in the indicated interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the interval containing the zero into 10 subintervals.
f (x) = x^3 − 4x + 2; interval: (1, 2)
I don't understand the instructions. How is this done?
Do you know what the "Intermediate Value Theorem" is?
f(1)= 1- 4+ 3= -1 and f(2)= 8- 8+ 2= 2. The Intermediate Value Theorem says that since this polynomial is continuous and positive at one end of the interval and negative at the other end there must be some point in the interval where f(x)= 0.
Now, divide (1, 2) into 10 subintervals:
(1, 1.1), (1.1, 1.2), (1.2, 1.3), (1.3, 1.4), (1.4, 1.5), (1.5, 1.6), (1.6, 1.7), (1.7, 1.8), (1.8, 1.9), and (1.9, 2.0).
Evaluate f(x) at the endpoints of those. If there is an interval where f(x) has different signs at the endpoints then, by the "Intermediate Value Theorem", f(x)= 0 somewhere in that interval. Divide that interval into 10 more subintervals and repeat. That will get you to things like 1.a0, 1.a1, 1,a2, etc, two decimal places. Repeat one more time to get three decimal places.