Question

Use the Intermediate Value Theorem to choose an interval over which the function, f(x)=-2x^3-3x+5, is guaranteed to have a zero.
a. [-3,-2]
b. [-2,0]
c. [0,2]
d. [2,4]

Answer 1

@mathmate can you help me?

Answer 2

From Wiki:
`In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.`

This means that if you can find two points x1, x2 on a continuous function such that f(x1)*f(x2) <0, which is another way of saying that f(x1) and f(x2) have different signs, then the Intermediate value theorem guarantees that there is a value x0 between x1 and x2 such that f(x0)=0, i.e. a zero of the function.
Example:

For f(x)=x^2-6x+8
we have x(3)=-1 (<0)
and x(0)=8 (>0)
so we know that there is a zero where the graph crosses the x-axis. In general, we do not know where this occurs, but we know that it does.
In this particular case, we have x=2, or f(2)=0.