Determine between which consecutive integers one or more real zeros of f(x) = –x3 + 2x2 – x – 5 are located

What method are you supposed to use.

Your cubic has one real root (a zero) at
x = appr. -1.116 plus 2 imaginary roots.

So I guess a real root lies between -1 and -2

check:
f(-1) = -1 + 2 + 1 - 5 = -3
f(-2) = 8 + 8 + 2 -5 = + 13

so the curve must have crossed the x-axis (the zero) somewhere between - 1 and -2

Where you supposed to try different f(x) 's ?

A third degree equation? I think I would graph it on your graphing calculator, see where it crosses the x axis.

Bob,

check my reply to your email from yesterday.
It applies to this question.

To determine between which consecutive integers one or more real zeros of the function f(x) = -x^3 + 2x^2 - x - 5 are located, you can use the intermediate value theorem.

The intermediate value theorem states that if a continuous function changes sign between two points, then it must have at least one zero between those points.

In this case, the function f(x) is a polynomial of degree 3, so it is continuous over the entire real number line.

To apply the intermediate value theorem, you can evaluate the function at various integer values and look for sign changes.

Let's evaluate the function at a few consecutive integers:

For x = -2: f(-2) = -(-2)^3 + 2(-2)^2 - (-2) - 5 = -8 + 8 + 2 - 5 = -3
For x = -1: f(-1) = -(-1)^3 + 2(-1)^2 - (-1) - 5 = -1 + 2 + 1 - 5 = -3
For x = 0: f(0) = -(0)^3 + 2(0)^2 - (0) - 5 = 0 + 0 + 0 - 5 = -5
For x = 1: f(1) = -(1)^3 + 2(1)^2 - (1) - 5 = -1 + 2 - 1 - 5 = -5
For x = 2: f(2) = -(2)^3 + 2(2)^2 - (2) - 5 = -8 + 8 - 2 - 5 = -7

From the evaluations, we can see that the function changes sign between x = -2 and x = -1 (from negative to negative), and also between x = 0 and x = 1 (from negative to negative). This means that there must be at least one real zero between each pair of consecutive integers: -2 and -1, and 0 and 1.

Therefore, one or more real zeros of the function f(x) = -x^3 + 2x^2 - x - 5 are located between -2 and -1, and between 0 and 1.