2b. Explain how the Intermediate Value Theorem (IVT)can be used

Answered: Explain how the Intermediate Value…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

2b.

Explain how the Intermediate Value Theorem (IVT)
can be used

Expert Solution

Step 1

Intermediate value theorem is stated as follows.

Consider an interval $I = [a, b]$ of real numbers $ℝ$ and a continuous function $f : I \to ℝ$ . If $u$ is a number between $f (a)$ and $f (b)$ then there is a $c \in (a, b)$ such that $f (c) = u$ .

According to the question, we are interested in the interval $I = [3, 4]$ .

$f (x) = k (x) + \cos (\frac{π}{2} x) - \sqrt{3}$

Note that $k (x)$ is given to be a continuous function, the cosine function is known to be a continuous function, and a constant function is a continuous function. As the addition of continuous functions is continuous, $f (x)$ is a continuous function (as required in the statement of the intermediate value theorem).

We need to find the root of $f (x)$ . So, take $u = 0$ .

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