International Mathematics Competition 2020 Day 2 Problem 1 | A Derivative Inequality

This video discusses an interesting differential inequality that appeared on the 2020 International Mathematics Competition. It showcases both how to develop intuition for inequalities involving derivatives, and how to interpret the results from these geometrically.

#Inequalities #MathematicsCompetitions #DifferentialCalculus

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[Szynkaa]

I was about to say the solution is wrong, but then i noticed i considered this problem with domain as real POSITIVE numbers, and i found whole class of functions satysfying this, for example f(x)=1/x (it even makes inequality become equality)

[thedoublehelix]

wow I would've never thought of that!

[hrs]

Definitely only positive constants are allowed
Btw really nice solution, I often overlook at the geometry of curves of the functions
I think this can be generalised looking at the convexity of functions and whether they are positive / negative on R

[math]

understand the process prof, great job

[wesleydeng]

Prof. Omar, since f(x)>0 and f'(x)^2>=0, from the inequality we can say f''(x)>=0. The rest of analysis will be similar. This way, we don't have to introduce g(x).

[rounaksinha]

Thanks, sir, as always very nice video.

[mikelolis]

I was genuinely just looking at this problem thirty minutes ago and trying to solve it lmao

[cyberman]

Messed up whole solution, this is first derivative of ( f(x)/f'(x)) after rearranging terms in the given equation.

[MyOneFiftiethOfADollar]

f(x)=x^3 gives 6x(x^3) < 2(3x^2)^2 = 18x^4 on the interval (1, inf) as a get "your feet wet" way of discovering the type of functions that may or may not satisfy the derivative based inequality.

Thx for interesting topic, but always disappointing when the solution is a constant. Would be much more interesting imo if problem yielded "non trivial" solutions with clean way to describe the the solution.

[rounaksinha]

Sir also please bless me, I am going to start my math major very shortly. I am very excited!

[rounaksinha]

Sir, please help me in proving Bonse inequality. I tried the proof by induction but that didn't work. Sir is it possible that the gap between the two consecutive primes become so large that the square of the latter exceeds the product of preceding primes.