Intermediate Value Theorem Explained - To Find Zeros, Roots or C value - Calculus

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Imagine a question asking you to determine that f(x)=1/n+1 admits one unique solution Un - being a progression - and determining the expression of Un!

Théorème des valeurs intermidiaires

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Assalamualikum sir. ! I have an issue kindly help me if we had given a polynomial of highest degree of cube 4x^3-6x^2+3x-2 and the closed interval is [1,2] ! When we plug in 1 f(1) =-1 and when we plug in f(2)=12 now in order to find f(c) we will equate it by setting equal to zero but how the polynomial of degree 3 will be solved? How to find the value of C ?

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I have a question, why did you curve the graph like a wave instead of straight line or a U shape?

On the first example x=-4 and x=3

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For the first example, you said that c has to be = to 0 but then in the second example c is 19 and never passes through the x-axis. Is this because the first question specifies finding the "root" while the second question doesn't?

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I'm confused where you say that -3 is not btn -6 & 8 🤔🤔🤔
Can you explain there

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why we said 19

In example 2 f(1) and f(4) both are positive sign ..while we say in intermediate value theorem must be different signs

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excellent explanation!

To test the c value you can also plug it into the original equation. So

19 = 2(2)^2 + 3(2) + 5 => 19 = 19