A Very Nice Exponential Equation | Best Solution To Exponential Challenge | Hints To Algebraic Maths

A Very Nice Exponential Equation/Best Solution To Exponential Challenge is a mathematical video you will ever love to watch because there are many tricks and hints you will definitely learn from this video clip.
In this video, I will show you how you can solve mathematical challenges of this kind without difficulties.
Watch from the beginning to the end without skipping any parts in order to get the full knowledge to solving this challenge and seminar mathematical challenges you will/may come across in the world of mathematics.
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[petersonmatos]

greatful solution sir... what a great professor 👏👏

[matheusnuci]

hello sir, my private mathematics teacher recommended your channel to me, I loved the work and thank you for the teachings, and I look forward to new learnings

[rottenturnip]

First of all it is reasonable that x has to positive. If it’s negative then between x and x^3, one has to out-negate the positiveness of x^2. We can do the same with 1/x, 1/x^2, 1/x^3, to see that x being negative makes the left hand side negative. Therefore x is positive and by AM-GM (arithmetic mean geometric mean inequality) we have that x+1/x geq 2, equality iff x=1. We can do the same with x^2+1/x^2 and x^3+1/x^3 to see that our initial equality is the equality case of all three AM-GM applications. This only happens if x=1 and we can see that this works.

[ericluz]

Sir, we could solve thia question simply by observing the inequality between arithmetic and geometric averages.

x+(1/x)≥2, since a+b≥2√ab
x²+(1/x²)≥2
x³+1/x³≥2

If we sum these equations, we get that the expression shown in the exercise is always ≥6. But it is equal six if, and only if, x=1, which leads us to the solution.

[zakiabg]

Is there a general formula to solve n degree equation, bro?

[nagarajahshiremagalore]

Time optimization is very poor while solving the problem.

[closer_to_the_unknown]

Let's estimate the value on the left. Using inequalities about averages, we have that x + 1/x >= 2 for positive x. By analogy, x^2 + 1/x^2 >= 2 and x^3 +1/x^3 >= 2, that is, the minimum value of the left side of the equation cannot be less than 6. Equality is achieved only when x = 1