Proof of existence by I.V.T.

I used the Sign Rule (I think that's the name) to show that there is exactly one solution that's negative. As originally stated, the number the problem asks for is a solution of x = x^3 + 3, which works out to x^3 - x + 3 = 0. If x is a solution, let y = -x. Then -y^3 + y + 3 = 0. The polynomial has exactly one sign change, so it has exactly one positive zero, but if y is positive, then x, which is -y, is negative.

ETA: Also, I find 10 and -10 to be easier to use than smaller numbers (other than 1). My test here would be (-10)^3 - (-10) + 3 = -1000 + 100 + 3 << 0

Nearest answer is - 1.672, it is still an approximate.
Question remains: is there an exact solution ?
Probably not in rational numbers, but may be an irrational one.
Oh, That is why they are called irrational number.

Your videos are great, sir.
I really appreciate them.
Best wishes to you.

I love this channel

??? By observation — Every cubic (odd polynomial as well) has at least one real solution. Plus there is a cubic “formula” despite its modern perceived complexity — especially when the x^2 term is 0 which was first solved before the generalized cubic.

I like Your English very much!

depressed cubics

nice as usual, please try to change your blackboard to a white one or improve the light system for better visualization of your videos, best regards👍

Solution:



is there a real solution to x = x³ + 3?
x = x³ + 3 |-x³
x - x³ = 3 |therefore x³ < x, which means that, as the result is an integer, x has to be negative!
(x³ < x would also be possible, if 0 < x < 1, but then the result of x - x³ would not be an integer)

Since x³ is growing very fast, x has to be quite small.

testing left term assuming x = -1
-1 - (-1)³ = -1 - (-1) = -1 + 1 = 0

testing left term assuming x = -2
-2 - (-2)³ = -2 - (-8) = -2 + 8 = 6

Therefore there is a real solution of x between -1 and -2.

Wow

U soooo intelligent

Thank you for saving my ass

Thanks for the help!!

It is not so difficult to calculate x
Assume that x is sum of two unknowns,plug in into the equation
use binomial expansion , rewrite as system of equations
Transform this system of equations to Vieta formulas for quadratic
Check if solution of quadratic satisfies system of equations before transformation

Isn't that also called Bolzano's Theorem?

... A good day to you Newton despite the bad weather, Didn't I tell you ... BLACKBOARD -> SIR. NEWTON <- PRESENTATIONS ... is your trademark, don't forget this! Even MIT still uses this way to teach, no digital devices needed ... By the way my friend, great clear application example of the I.V.T. Thank you again Newton, Jan-W

Awesome demonstration of the use of the IVT.

Thanks sir . Please make a video on mid term and final exam reviews calculus 1