Unusual Induction Inequality Proof (1 of 3: Base case)

[mynanic]

thanks Eddie, you have supported the class of 2021 so much. i appreciate for all the help you have done for supporting us. us year 12 of 2021s wouldn’t be seeing you much anymore as we would be becoming bigger and better people. but I would thank you in the behalf of thousands of year 12s and 2022. may your career continue to being full of surprises and ups and downs. future students please be grateful for Eddie giving you math videos before exams or guidance in improving in maths 🙏

[whiteender]

Hands down the best math teacher I’ve ever seen. Truly brilliant. Keep up the fantastic work, you are what every educator should aspire to be.

[wasimvillidad]

For n = 1, considering LHS - RHS and showing algebraically that it is ≥ 0 seems more intuitive. AM-GM inequality as another option before calculus.

[沈博智-xy]

AM-GM could also be used to show that the base case a + 1/a >= a^0 + 1/a^0 = 1 + 1/1 = 2

LHS = a + 1/a

we know a > 0, so 1/a > 0, thus we can use AM-GM
LHS >= 2sqrt(a(1/a)) = 2sqrt(1) = 2 = RHS

[harryguanous]

Oddly enough, this was at the right time before my exams.

[michaelempeigne]

by AM - GM inequality

[ a + ( 1 / a ) ] / 2 > or = sqrt [ a * ( 1 / a ) ]
[ a + ( 1 / a ) ] > or = 2 * sqrt ( 1 )
a + ( 1 / a ) > or = 2
QED

[ARKGAMING]

I don't get why you'd use differential calculus when you could just consider a+1/a-2=(a²-2a+1)/a=(a-1)²/a
a is positive and real
Hence (a-1)² is non negative
Hence a + 1/a - 2 ≥ 0 (non negative divided by a positive is non negative)
a + 1/a ≥ 2

[SHM_]

Sir .... You're great. 😲

- From India

[maggiesun]

Hi Eddie, fellow maths/chem teacher coming by. Loved how these series of videos are filmed - very good for preping Year 11-12 students for exams by going through examples! I wonder if you could share a bit about what you used to film these series of videos? Namely, how does the set-up look like and which app/device for showing working out. I think it's important to still show students our faces but also take them through working out step by step. Zoom alone sometimes just doesn't cut it - haha~

[abhchow]

Instead of using calculus to prove that LHS ≥ RHS, I rearranged it as a quadratic.

Assume that a + 1/a ≥ 2
Then a^2 + 1 ≥ 2a
Then a^2 - 2a + 1 ≥ 0

Check the discriminant of the quadratic
Δ = 2^2 - 4*1*1 = 0
Therefore a^2 - 2a + 1 ≥ 0
Therefore a + 1/a ≥ 2

[samiyahossain]

Thank you for helping me so much! Your videos have been so useful.

[BobMingJr]

I think it would be easier to show that a + 1/a ≥ 2 by multiplying both sides by ‘a’ then factorising LHS-RHS≥0. eg.
a² + 1 ≥ 2a
a² - 2a + 1 ≥ 0
(a-1)² ≥ 0
since a is real, (a-1)² must be equal to or greater than 0

[mahbubhassan]

Hi Eddie, Could you please share how you make this kind of videos using Ipad and computer.?

[fareedabifarraj]

@Eddie Woo
What kind of drawing pad are you using?
I would love to buy a similar one for my online teaching
And can you please tell me what whiteboard application are you using?

[gordonli]

Why do you even need to use calculus for that proof. You could have substitute a = 1 in tats = 2. And if a>=2 it must be greater than RHS. QED.

[qweboss]

Can you tell (√x)^2 and √(x^2) this both are same ?
I think no because, (√x) output is whether plus or minus x but power it with 2 will converts into positive
And in second cause we are squareing x before rooting so when we root after square ing it will gives plus or minus x .
That's what I am thinking can anyone correct me