use interediate value theoremo prove that 2cosk -3x=Oa- real zerotobetween O and I

Answered: use interediate value theorem o prove…

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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use interediate value theorem
o prove that 2cosk -3x=O
a- real zero
to
between O and I

Expert Solution

Step 1

Hey, since there are multiple questions posted, we will answer first question. If you want any specific question to be answered then please submit that question only or specify the question number in your message.

Note that, the Intermediate Value theorem states that, "For two real numbers a and b with a < b, let f be continuous function on closed interval [a, b]. Then for every c between f(a) and f(b) there exist m belongs to [a, b] such that, f(m)=c."

Step 2

Consider the given function $f (x) = 2 \cos (x) - 3 (x)$ .

Since every cosine function is continuous on R, the function $f (x) = 2 \cos (x) - 3 (x)$ is continuous on $[0, 1]$ .

Evaluate the value of the function $f (x) = 2 \cos (x) - 3 (x)$ at the endpoints of the interval $[0, 1]$ as follows.

$\begin{array}{rcl} f (0) & = & 2 \cos (0) - 3 (0) \\ = & 2 (1) - 0 \\ = & 2 > 0 [p o s i t i v e] \\ f (1) & = & 2 \cos (1) - 3 (1) \\ = & 2 \cos (1) - 3 \\ = & - 1.91939 < 0 [n e g a t i v e] \end{array}$

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