Question

three conditions of rolles thereom is exact or sufficient or necessary?
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Answer 1

ᴄᴏʀʀᴇᴄᴛ ǫᴜᴇsᴛɪᴏɴ :-

ᴛʜʀᴇᴇ ᴄᴏɴᴅɪᴛɪᴏɴs ᴏғ ʀᴏʟʟᴇs ᴛʜᴇʀᴇᴏᴍ ɪs ᴇxᴀᴄᴛ ᴏʀ sᴜғғɪᴄɪᴇɴᴛ ᴏʀ ɴᴇᴄᴇssᴀʀʏ?

ᴄᴏʀʀᴇᴄᴛ ᴀɴsᴡᴇʀ :-

ᴀʟʟ ᴛʜʀᴇᴇ ᴄᴏɴᴅɪᴛɪᴏɴs ᴏғ ʀᴏʟʟᴇ's ᴛʜᴇᴏʀᴇᴍ ᴀʀᴇ ɴᴇᴄᴇssᴀʀʏ ғᴏʀ ᴛʜᴇ ᴛʜᴇᴏʀᴇᴍ ᴛᴏ ʙᴇ ᴛʀᴜᴇ: ғ(x) ɪs ᴄᴏɴᴛɪɴᴜᴏᴜs ᴏɴ ᴛʜᴇ ᴄʟᴏsᴇᴅ ɪɴᴛᴇʀᴠᴀʟ [ᴀ,ʙ]; ғ(x) ɪs ᴅɪғғᴇʀᴇɴᴛɪᴀʙʟᴇ ᴏɴ ᴛʜᴇ ᴏᴘᴇɴ ɪɴᴛᴇʀᴠᴀʟ (ᴀ,ʙ); ғ(ᴀ)=ғ(ʙ).

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs

ᴀɴsᴡᴇʀᴇᴅ ʙʏ :-

sᴀᴠᴀɢᴇ_ʙᴀʙᴇ⚡

Answer 2

ᴄᴏʀʀᴇᴄᴛ ǫᴜᴇsᴛɪᴏɴ :-

ᴛʜʀᴇᴇ ᴄᴏɴᴅɪᴛɪᴏɴs ᴏғ ʀᴏʟʟᴇs ᴛʜᴇʀᴇᴏᴍ ɪs ᴇxᴀᴄᴛ ᴏʀ sᴜғғɪᴄɪᴇɴᴛ ᴏʀ ɴᴇᴄᴇssᴀʀʏ?

ᴄᴏʀʀᴇᴄᴛ ᴀɴsᴡᴇʀ :-

ᴀʟʟ ᴛʜʀᴇᴇ ᴄᴏɴᴅɪᴛɪᴏɴs ᴏғ ʀᴏʟʟᴇ's ᴛʜᴇᴏʀᴇᴍ ᴀʀᴇ ɴᴇᴄᴇssᴀʀʏ ғᴏʀ ᴛʜᴇ ᴛʜᴇᴏʀᴇᴍ ᴛᴏ ʙᴇ ᴛʀᴜᴇ: ғ(x) ɪs ᴄᴏɴᴛɪɴᴜᴏᴜs ᴏɴ ᴛʜᴇ ᴄʟᴏsᴇᴅ ɪɴᴛᴇʀᴠᴀʟ [ᴀ,ʙ]; ғ(x) ɪs ᴅɪғғᴇʀᴇɴᴛɪᴀʙʟᴇ ᴏɴ ᴛʜᴇ ᴏᴘᴇɴ ɪɴᴛᴇʀᴠᴀʟ (ᴀ,ʙ); ғ(ᴀ)=ғ(ʙ).

ᴀɴsᴡᴇʀᴇᴅ ʙʏ :-

ᴘɪɴᴋ_ᴡɪɴᴇ⚡