How do you use the intermediate value theorem to show that there is a root of the equation 2x³+ x²+ 2 = 0 over the interval (-2, -1)?

Solution:

Intermediate Value Property makes functions that are continuous on interval particularly useful in mathematics and its applications. A function is said to have the Intermediate Value Property if whenever it takes two values, it also takes all the values in between.

Mathematically it is called the Intermediate Value Theorem for Continuous Functions.

A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). In other words, if \(y_{o}\) is any value between f(a) and f(b), then \(y_{o}\) = f(c) for some c [a,b].

The function given in the problem statement i.e. 2x³+ x² + 2 = 0 is plotted in the function below.

The above shows that the zero of the function is called the solution of the function. The Intermediate Value Theorem tells us that if f is continuous, then any interval on which the sign changes contains a zero of the function.

In the above graph, it is apparent that the function becomes zero when the function changes from -10 to 1. So we then focus on the interval between y = -10 and y = 1.

Now comes the step to determine the value of x at which y = 0. We draw another graph to come as close to the point where y = 0.

The point where y is almost zero is x = -1.197 which in other words is zero of the function. Hence x = -1.197 lies between the interval [-2, -1].