Show that the cubic p(x)=x^3+a x^2+b x+c has extreme values...

Key Concepts

Cubic Polynomial

A cubic polynomial is a third-degree polynomial of the form p(x) = x³ + ax² + bx + c. It is a fundamental element in algebra and calculus that can exhibit various behaviors, including inflection points and concurrent extreme values (maximum and minimum). The coefficients define its shape and the nature of its roots and extrema.

Derivative and Critical Points

Critical points of a function occur where its first derivative is zero or undefined. For a cubic polynomial, the computation of the first derivative leads to a quadratic equation. The solutions to this quadratic correspond to the potential local maximum and minimum of the original cubic function, marking the locations where its rate of change is zero.

Discriminant of a Quadratic

The discriminant of a quadratic equation, found in the expression for the roots of a quadratic, determines the number and nature of its solutions. For the derivative of a cubic polynomial, a positive discriminant indicates two distinct real critical points, which correspond to the existence of both a local maximum and minimum, hence showing extreme values.

Extreme Value Criterion

In the context of cubic polynomials, having extreme values is tied to the characteristics of its derivative. Specifically, since the derivative is quadratic, the condition for two real critical points is that its discriminant must be positive. In the given problem, this condition translates to a relationship between the coefficients, namely a² > 3b, establishing the criterion for the existence of extreme values.

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