Critical Numbers Consider the cubic function f(x)=a x^3+b x^2+c x+d, where a ≠0 . Show that f ca

step 1 So our question is that we have to show that the general cubic function can have either 0, 1, or

Critical Numbers Consider the cubic function f(x)=a x^3+b...

Key Concepts

Discriminant

The discriminant of a quadratic equation, given by b² - 4ac, determines the quantity and nature of the equation's real roots. In the context of finding critical numbers for a cubic function, the discriminant of the derivative (which is quadratic) is used to ascertain whether there are two distinct critical numbers (positive discriminant), a single repeated critical number (zero discriminant), or no real critical numbers (negative discriminant).

Critical Numbers

Critical numbers are values in the domain of a function at which the derivative is zero or undefined. These points are significant because they are potential indicators of local extrema or saddle points. For smooth functions like polynomials, the search for critical numbers is concentrated on the points where the derivative equals zero.

Quadratic Equation

When the derivative of a cubic function is taken, the result is a quadratic equation. This quadratic reflects the changes in the slope of the original cubic function, and solving it reveals the potential critical numbers. The number and nature of the solutions of this quadratic thus directly determine the number of critical numbers the cubic function may have.

Derivative

The derivative of a function represents its instantaneous rate of change and is fundamental in calculus for analyzing the behavior of functions. For polynomial functions, the derivative is obtained by differentiating each term, resulting in a lower-degree polynomial which provides information about the slopes of the original function, such as where it is increasing or decreasing.

Cubic Function

A cubic function is a third-degree polynomial typically expressed in the form f(x) = ax³ + bx² + cx + d, where a ? 0. These functions are notable for their distinctive S-shaped curves and can have various behaviors including local maxima and minima as well as points of inflection, all of which are closely linked to the function's critical points.

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