Question
Use the intermediate theorem to show that x^3+x+3=0 has a solution in interval (-2, 0)
216
likes1079 views
Answer to a math question Use the intermediate theorem to show that x^3+x+3=0 has a solution in interval (-2, 0)
Andrea
4.583 Answers
To apply the Intermediate Value Theorem, we need to ensure that the function is continuous on the given interval and also that the function takes on both positive and negative values on that interval.
First, let's check the continuity of the function f(x) = x^3 + x + 3 on the interval (-2, 0). The function is a polynomial, and polynomials are continuous everywhere. Therefore, f(x) is continuous on (-2, 0).
Next, let's evaluate the function at the endpoints of the interval to check if it takes on both positive and negative values.
f(-2) = (-2)^3 + (-2) + 3 = -2
f(0) = (0)^3 + (0) + 3 = 3
Since f(-2) = -2 and f(0) = 3, the function f(x) takes on both positive and negative values on the interval (-2, 0).
Therefore, by the Intermediate Value Theorem, there exists a number c in the interval (-2, 0) such that f(c) = 0. This means that the equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).
Answer: The equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).
First, let's check the continuity of the function f(x) = x^3 + x + 3 on the interval (-2, 0). The function is a polynomial, and polynomials are continuous everywhere. Therefore, f(x) is continuous on (-2, 0).
Next, let's evaluate the function at the endpoints of the interval to check if it takes on both positive and negative values.
f(-2) = (-2)^3 + (-2) + 3 = -2
f(0) = (0)^3 + (0) + 3 = 3
Since f(-2) = -2 and f(0) = 3, the function f(x) takes on both positive and negative values on the interval (-2, 0).
Therefore, by the Intermediate Value Theorem, there exists a number c in the interval (-2, 0) such that f(c) = 0. This means that the equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).
Answer: The equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).
Frequently asked questions (FAQs)
Math question: How many different types of triangles can be formed if the lengths of their sides do not exceed 10 units?
+
How many ways can a committee of 5 people be selected from a group of 10 candidates?
+
Math question: In a triangle with side lengths A = 5, B = 7, and angle C = 45 degrees, find the length of side C using the Sine Law.
+
New questions in Mathematics