Mathematics Problem: Function Variations and Tangent Equation
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Mathematics, function variations, tangent equation, differentiability, polynomial functions, rational functions, intermediate value theorem
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Abstract
This document provides a detailed analysis of the function g(x) = x^3 - 3x - 4 and f(x) = (x^3 + 2x^2) / (x^2 - 1), including their differentiability, sign, and variation tables, as well as the equation of the tangent to the curve C at x = 2.
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[...] is 10?2 near using the calculator: ??2.20 The sign of the function g on R : on and on 2. We consider the function f defined on by f(x)=(x3+2x2)/(x2?1). The function f is differentiable on since it is a rational function, and On the interval x>0 and so the sign of is the sign of g(x). Thus, the function f is decreasing on and increasing on The variation table of f on : C ) To perform the calculation, I have replaced the approximate value of ? [...]
[...] We consider the function g defined on R by g(x)=x3?3x?4 The function g is differentiable on R since it is a polynomial, and vanishes at 1 and and is of the sign of a=3>0 for values outside the roots. De plus, And and From which the variation table on On the interval the function g reaches a maximum equal to so the equation has no solution on this interval. On the interval the function g is continuous because it is a polynomial, and g is strictly increasing on Therefore, by the intermediate value theorem, the equation has a unique solution ? on and therefore in R. An approximate value of ? [...]
