Question

Use the Intermediate Value Theorem to choose the intervals over which the function, 2x^3 - 3x + 5, is guaranteed to have a zero.

Answer 1

We know that, as per a corollary of intermediate value theorem, if a function f(x) is continuous on a closed interval [a,b], and values of f(a) and f(b) have opposite signs, then the function f(x) is guaranteed to have a zero on the interval (a,b).

So, basically, we need to figure out two values of x, at which the values of the given cubic function have opposite signs.

Let us consider the interval [-2,1].

We have $f(x)=2x^{3}-3x+5$. Upon substituting the values x=-2 and x=1 one by one, we get:

$f(x)=2(-2)^{3}-3(-2)+5=-16+6+5=-5$

$f(x)=2(1)^{3}-3(1)+5=2-3+5=4$

We can see that signs of values of the function at x=-2 and x=1 are opposite, therefore, as per intermediate value theorem, the function is guaranteed to have a zero on the interval [-2,1]