Question

define Lagrange mean value theorem?

Answer 1

Lagrange’s mean value theorem is the mean value theorem itself or also called first mean value theorem. Here in this article, we will learn both the theorems. By mean we understand the average of the given values. But in the case of integrals, the process of finding the mean value of two different functions is different. Let us learn the mean value of such functions along with their geometrical interpretation.

Lagrange’s Mean Value Theorem

If a function f is defined on the closed interval [a,b] satisfying the following conditions –

i) The function f is continuous on the closed interval [a, b]

ii)The function f is differentiable on the open interval (a, b)

Then there exists a value x = c in such a way that

f'(c) = [f(b) – f(a)]/(b-a)

This theorem is also known as the first mean value theorem or Lagrange’s mean value theorem.

Answer 2

Answer:

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ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ ɪs ᴏɴᴇ ᴏғ ᴛʜᴇ ᴄᴇɴᴛʀᴀʟ ᴛʜᴇᴏʀᴇᴍs ᴏғ ᴀʙsᴛʀᴀᴄᴛ ᴀʟɢᴇʙʀᴀ. ɪᴛ sᴛᴀᴛᴇs ᴛʜᴀᴛ ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ғᴏʀ ᴀɴʏ ғɪɴɪᴛᴇ ɢʀᴏᴜᴘ sᴀʏ ɢ, ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ sᴜʙɢʀᴏᴜᴘ ʜ ᴏғ ɢʀᴏᴜᴘ ɢ ᴅɪᴠɪᴅᴇs ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ɢ. ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɢʀᴏᴜᴘ ʀᴇᴘʀᴇsᴇɴᴛs ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ᴇʟᴇᴍᴇɴᴛs. ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ ᴡᴀs ɢɪᴠᴇɴ ʙʏ ᴊᴏsᴇᴘʜ-ʟᴏᴜɪs ʟᴀɢʀᴀɴɢᴇ. ɪɴ ᴛʜɪs ᴀʀᴛɪᴄʟᴇ, ʟᴇᴛ ᴜs ᴅɪsᴄᴜss ᴛʜᴇ sᴛᴀᴛᴇᴍᴇɴᴛ ᴀɴᴅ ᴘʀᴏᴏғ ᴏғ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ᴀɴᴅ ᴀʟsᴏ ʟᴇᴛ ᴜs ʜᴀᴠᴇ ᴀ ʟᴏᴏᴋ ᴀᴛ ᴛʜᴇ ᴛʜʀᴇᴇ ʟᴇᴍᴍᴀs ᴜsᴇᴅ ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ ᴡɪᴛʜ ᴛʜᴇ ᴇxᴀᴍᴘʟᴇs. sᴇᴛ ᴛʜᴇᴏʀʏ sᴇᴛs ᴍᴀᴛʜs ғɪɴɪᴛᴇ ᴀɴᴅ ɪɴғɪɴɪᴛᴇ sᴇᴛs sᴜʙsᴇᴛ ᴀɴᴅ sᴜᴘᴇʀsᴇᴛ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ sᴛᴀᴛᴇᴍᴇɴᴛ ᴀs ᴘᴇʀ ᴛʜᴇ sᴛᴀᴛᴇᴍᴇɴᴛ, ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ sᴜʙɢʀᴏᴜᴘ ʜ ᴅɪᴠɪᴅᴇs ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɢʀᴏᴜᴘ ɢ. ᴛʜɪs ᴄᴀɴ ʙᴇ ʀᴇᴘʀᴇsᴇɴᴛᴇᴅ ᴀs; |ɢ| = |ʜ| ʙᴇғᴏʀᴇ ᴘʀᴏᴠɪɴɢ ᴛʜᴇ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ, ʟᴇᴛ ᴜs ᴅɪsᴄᴜss ᴛʜᴇ ɪᴍᴘᴏʀᴛᴀɴᴛ ᴛᴇʀᴍɪɴᴏʟᴏɢɪᴇs ᴀɴᴅ ᴛʜʀᴇᴇ ʟᴇᴍᴍᴀs ᴛʜᴀᴛ ʜᴇʟᴘ ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜɪs ᴛʜᴇᴏʀᴇᴍ. ᴡʜᴀᴛ ɪs ᴄᴏsᴇᴛ? ɪɴ ɢʀᴏᴜᴘ ᴛʜᴇᴏʀʏ, ɪғ ɢ ɪs ᴀ ғɪɴɪᴛᴇ ɢʀᴏᴜᴘ, ᴀɴᴅ ʜ ɪs ᴀ sᴜʙɢʀᴏᴜᴘ ᴏғ ɢ, ᴀɴᴅ ɪғ ɢ ɪs ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ, ᴛʜᴇɴ; ɢʜ = { ɢʜ: ʜ ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ʜ } ɪs ᴛʜᴇ ʟᴇғᴛ ᴄᴏsᴇᴛ ᴏғ ʜ ɪɴ ɢ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ ᴀɴᴅ ʜɢ = { ʜɢ: ʜ ᴀɴ ᴇʟᴇᴍᴇɴᴛ ᴏғ ʜ } ɪs ᴛʜᴇ ʀɪɢʜᴛ ᴄᴏsᴇᴛ ᴏғ ʜ ɪɴ ɢ ᴡɪᴛʜ ʀᴇsᴘᴇᴄᴛ ᴛᴏ ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛ ᴏғ ɢ. ɴᴏᴡ, ʟᴇᴛ ᴜs ʜᴀᴠᴇ ᴀ ᴅɪsᴄᴜssɪᴏɴ ᴀʙᴏᴜᴛ ᴛʜᴇ ʟᴇᴍᴍᴀs ᴛʜᴀᴛ ʜᴇʟᴘs ᴛᴏ ᴘʀᴏᴠᴇ ᴛʜᴇ ʟᴀɢʀᴀɴɢᴇ ᴛʜᴇᴏʀᴇᴍ. ʟᴇᴍᴍᴀ : ɪғ ɢ ɪs ᴀ ɢʀᴏᴜᴘ ᴡɪᴛʜ sᴜʙɢʀᴏᴜᴘ ʜ, ᴛʜᴇɴ ᴛʜᴇʀᴇ ɪs ᴀ ᴏɴᴇ ᴛᴏ ᴏɴᴇ ᴄᴏʀʀᴇsᴘᴏɴᴅᴇɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ʜ ᴀɴᴅ ᴀɴʏ ᴄᴏsᴇᴛ ᴏғ ʜ. ʟᴇᴍᴍᴀ : ɪғ ɢ ɪs ᴀ ɢʀᴏᴜᴘ ᴡɪᴛʜ sᴜʙɢʀᴏᴜᴘ ʜ, ᴛʜᴇɴ ᴛʜᴇ ʟᴇғᴛ ᴄᴏsᴇᴛ ʀᴇʟᴀᴛɪᴏɴ, ɢ ∼ ɢ ɪғ ᴀɴᴅ ᴏɴʟʏ ɪғ ɢ ∗ ʜ = ɢ ∗ ʜ ɪs ᴀɴ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ʀᴇʟᴀᴛɪᴏɴ. ʟᴇᴍᴍᴀ : ʟᴇᴛ s ʙᴇ ᴀ sᴇᴛ ᴀɴᴅ ∼ ʙᴇ ᴀɴ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ʀᴇʟᴀᴛɪᴏɴ ᴏɴ s. ɪғ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴛᴡᴏ ᴇǫᴜɪᴠᴀʟᴇɴᴄᴇ ᴄʟᴀssᴇs ᴡɪᴛʜ ᴀ ∩ ʙ = ∅, ᴛʜᴇɴ ᴀ = ʙ.

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