Finding Solutions Using the Intermediate Value Theorem

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In summary, the conversation discusses how to show that the equation e^x = x + 2 has exactly two solutions. The person suggests using the intermediate value theorem and creating a function to prove the solution. However, they are unsure about the boundaries and ask for advice on how to proceed.
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Vashti
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Homework Statement



Show that the equation e^x = x + 2 had exactly two solutions. Explain your reasoning.

Homework Equations



Okay so I guess I'm using the intermediate value theorum and proving that f(a)f(b) < 0 we can find a c and d such that f(c) = 0 and f(d) = 0 so that by collorary it's true. This is an issue because I don't have boundaries for my function...

The Attempt at a Solution



create a function: f(x) = e^x - x - 2
and that's about as far as I got. I'm assuming I arbitarily select an interval and use that? Are there rule that I should follow though?
 
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  • #2
I would believe you should select some values and check for a sign change...
 

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a fundamental theorem in calculus that states that if a continuous function f(x) is evaluated at two points, a and b, and f(a) and f(b) have opposite signs, then there exists at least one point c between a and b where f(c) = 0.

Why is the Intermediate Value Theorem important?

The Intermediate Value Theorem is important because it guarantees the existence of a solution to equations that cannot be solved algebraically. It also provides a way to prove the existence of roots for polynomial functions, which is essential in many areas of mathematics.

What is the difference between the Intermediate Value Theorem and the Mean Value Theorem?

While both the Intermediate Value Theorem and the Mean Value Theorem are fundamental theorems in calculus, they serve different purposes. The Intermediate Value Theorem guarantees the existence of a solution to an equation, while the Mean Value Theorem provides a way to find the average rate of change of a function over a specific interval.

Can the Intermediate Value Theorem be applied to non-continuous functions?

No, the Intermediate Value Theorem only applies to continuous functions. A function is considered continuous if there are no sudden jumps or breaks in its graph.

How is the Intermediate Value Theorem used in real life?

The Intermediate Value Theorem is used in many real-life applications, such as in physics and engineering, to prove the existence of certain values or solutions. It is also used in economics to prove the existence of equilibrium points in supply and demand curves.

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