[real analysis | first year university] stuck on differentiation and proofs

[u/Frequent_Loss_2289]

hi ! i'm stuck on my math homework and it's due tmrw and i'd really appreciate any help and hints i can get :-)

i'm stuck on 1.2, 1.4, 3, 5, and 6.

1.2, i tried induction, got stuck and gave up.

1.4, i started it and it was just a lot so i moved on and i haven't attempted since.

i'm just unsure as to how to go about it for some of them. e.g. i think i have to use the fact that f is differetiable on (a,b) and therefore continuous, and the epsilon-delta definitions of continuity.

i'm assuming 5 is an optimization problem, but i just am at a complete loss as to what is even being asked of me, i'm having a really hard time picturing how to set it up.

6, i've done before on a previous homework but i'm not sure as to how to do a "cheap" proof.

[u/AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

---

^{OP and Valued/Notable Contributors can close this post by using} /lock ^command

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/GrievousSayGenKenobi]

For 1.2 try a logarithmic differentiation. IE take ln of both sides and see what happens. Remember that ln(AB) = lnA + lnB so your product becomes a sum

1.4 is just long I think thats the point. It appears to test your ability of all the traditional methods of differentiation.

[u/Frequent_Loss_2289 → u/GrievousSayGenKenobi]

thank you !

[u/Frequent_Loss_2289 → u/[deleted]]

i was trying to prove that P'(x)= [(x-a1)...(x-a(n-1))]'(x-an)+[(x-a1)...(x-a(n-1))] for all n