Going Crazy over this Mean Value Theorem Question!!!

[kittie21]

Okay so the question is asking to show that for 0<x<y, sqrty - sqrtx < (y - x)/(2 sqrt x) (using the mean value theorem)

I think I have some valuable information pertaining to the question...but I am so confused about how to solve it I could just be pulling numbers out of nowhere in a desperate attempt to feel smart!

I have the slope of the tangent for sqrty - sqrt x = -sqrt y / y and the slope of the tangent for (y - x)/(2 sqrt x) = -(1/2y)/ y But not sure what I do with this information! any help would be amazingly appreciated!

[Martingale]

Okay so the question is asking to show that for 0<x<y, sqrty - sqrtx < (y - x)/(2 sqrt x) (using the mean value theorem)

I think I have some valuable information pertaining to the question...but I am so confused about how to solve it I could just be pulling numbers out of nowhere in a desperate attempt to feel smart!

I have the slope of the tangent for sqrty - sqrt x = -sqrt y / y and the slope of the tangent for (y - x)/(2 sqrt x) = -(1/2y)/ y But not sure what I do with this information! any help would be amazingly appreciated!

between and

let

...

[kittie21]

Thanks for the response! But I am not sure what I am supposed to do with this f(t) = sqrt t ... how does that fit in with the question? Am I using substitution with the t?

[Martingale]

Thanks for the response! But I am not sure what I am supposed to do with this f(t) = sqrt t ... how does that fit in with the question? Am I using substitution with the t?

between and

let

...

...

[kittie21]

...so.. the derivative of c, which is 1/2sqrt c should equal the slope of the tangent, sqrt y/ y. It makes sense in theory, but how do I make them equal eachother? Am I now supposed to solve for one of the variables? But there is two of them. I do not understand the next step. Thanks for your help so far though!

[Martingale]

...so.. the derivative of c, which is 1/2sqrt c should equal the slope of the tangent, sqrt y/ y. It makes sense in theory, but how do I make them equal eachother? Am I now supposed to solve for one of the variables? But there is two of them. I do not understand the next step. Thanks for your help so far though!

...

you want to show that

To get the above you need to answer for what value of is as large as possible...or
for what value of is as small as possible.

remember...

[kittie21]

Soo... can I designate numbers to the value of x and y as long as they fit in with 0<x<y? How do I know which numbers to use to do that?

[kittie21]

okay so, the f(y)-f(x)/y-x has to be greater than 0, which means that b-a > 0 which means that the divisor must be positive so that the answer stays above zero, so b>a! I don't need to find an exact answer, do I?!

[kittie21]

I mean , so y>x...but i just realized that that is what the 0<x<y was saying anyways. nevermind. I am still confused!

[Martingale]

okay so, the f(y)-f(x)/y-x has to be greater than 0, which means that b-a > 0 which means that the divisor must be positive so that the answer stays above zero, so b>a! I don't need to find an exact answer, do I?!

I'm not sure what you are trying to accomplish with the above post.


I think at this point, if you don't see how to finish, you should look in your book to see examples of how the Mean Value theorem can be used. If I give one more hint I'll be doing the entire problem. (which doesn't really help you)