Show that a cubic function (a third-degree polynomial)...

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three $x$ -intercepts $x_{1}, x_{2},$ and $x_{3},$ show that the $x$ -coordinate of the inflection point is $\left(x_{1}+x_{2}+x_{3}\right) / 3.$

Key Concepts

Cubic Functions

A cubic function, or third-degree polynomial, is a function of the form f(x) = ax^3 + bx^2 + cx + d where a is nonzero. Cubic functions are fundamental in algebra and calculus, exhibiting distinct characteristics such as a single inflection point and the possibility of up to three real roots. Their structure leads to unique behaviors in curvature and symmetry that are central to understanding their graphs.

Inflection Points

An inflection point on a curve is where the function changes its concavity – that is, it switches from being concave up to concave down or vice versa. For cubic functions, the inflection point is found by determining where the second derivative, which is a linear function in this case, equals zero. This guarantees that a cubic function always has exactly one point of inflection.

Differentiation and the Second Derivative Test

Differentiation, particularly the calculation of first and second derivatives, is a key concept in calculus for analyzing the behavior of functions. The second derivative is used to test for concavity and inflection points. In cubic functions, the simplicity of the second derivative (being linear) ensures there is a unique solution to the equation f''(x) = 0, thereby identifying a single inflection point.

Relationships Between Roots and Coefficients (Vieta's Formulas)

Vieta's formulas establish relationships between the roots of a polynomial and its coefficients. In the context of a cubic function, these relationships allow one to express the sum of its roots in terms of its coefficients. When analyzing the inflection point in relation to the x-intercepts, the x-coordinate of the inflection point can be shown to be the average of the roots, illustrating the interplay between a function's geometric properties and its algebraic representation.

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