Question

lnx=2-x^2
has exactly one root..

Answer 1

verify by calculation that the root lies between 1.0 and 1.4.

Answer 2

To verify this problem, bring this entire expression to one side e.g lnx + x^2 -2 = 0. From here, plug in x = 1 and x = 1.4. If the result of plugging them in leads to an opposite side answer, then you have verified that the root lies between 1.0 and 1.4. For example if x = 1 leads to the answer being less than zero and x = 4 leads to the answer being greater than zero than the root must like between them.

Answer 3

if i plug in x=1,
i get -1=0

Answer 4

I mean plugging x =1 without it equalling to 0 and the same for the other

Answer 5

@Hyper: You may have proved (by Intermediate Value Theorem) that there exists a root between x=1 and x=1.4 but that does not guarantee that there's no roots on the other intervals.

Answer 6

Yes true however the question just asks for that specific interval..

Answer 7

Stare at it again, and let g(x)=x62+ln(x)-2, you can see that it asks you to prove the uniqueness of the root (on the natural domain of the function g(x)) first, then it asks you to verify that the root actually lies in (1,1.4).

Answer 8

@drawar, I understand what you mean, but cannot find out that root

Answer 9

@kausarsalley : Have you heard of Mean Value Theorem?

Answer 10

@Loser66 : you don't need to find what the root is, just prove that it exists, and then prove it's unique.

Answer 11

no......

Answer 12

Which course are you taking @kausarsalley ?

Answer 13

And how about Intermediate Value Theorem?

Answer 14

so? need to take derivative to prove that there is only one critical point?

Answer 15

no..

Answer 16

@kausarsalley how can we solve the problem with all your "no" ???!!!

Answer 17

^ I second that, the two aforementioned theorems will act as effective weapons to solve this problem.

Answer 18

i am okay with hyper's explanations....

Answer 19

Fine, then don't say I didn't warn you it's wrong.

Answer 20

Basically Hyper's using the Intermediate Value Theorem explicitly, I don't figure out how you can grasp it without knowing of the theorem itself.

Answer 21

@drawar visualize it, quite simple, but how to put it in logic is the problem. I don't think Hyper's way is wrong, It's totally right. like