A cubic function is a polynomial of degree 3 ; that is, it...

A cubic function is a polynomial of degree $3 ;$ that is, it has the form $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a \neq 0$ . (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?

A cubic function is a polynomial of degree $3 ;$ that is, it has the form $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a \neq 0$ .
(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
(b) How many local extreme values can a cubic function have?

Key Concepts

Local Extreme Values

Local extreme values refer to the local maximums and minimums of a function, which occur at critical points where the derivative is zero and changes sign. For cubic functions, these are the possible peaks and valleys in the graph, and since the derivative is quadratic, a cubic function can have up to two local extreme values.

Critical Numbers

Critical numbers are the values of x where the derivative of a function is zero or does not exist. In the context of cubic functions, setting the quadratic derivative equal to zero allows us to find the critical numbers, which help in determining the locations of local extreme values or inflection points.

Discriminant Analysis

When the derivative of a cubic function is a quadratic equation, the discriminant (the part under the square root in the quadratic formula) determines the number of real roots. A positive discriminant yields two distinct critical numbers, a zero discriminant yields one repeated (double) critical number, and a negative discriminant means there are no real critical numbers.

Polynomial Degree

The degree of a polynomial, in this case three for a cubic function, indicates the highest power of x present in the function. This degree dictates key characteristics such as the maximum number of turning points and the end behavior of the function, which are central in understanding the graph’s overall structure.

Derivative

The derivative of a function measures its instantaneous rate of change and is instrumental in identifying where the function’s slope is zero or undefined. For a cubic function, taking the derivative leads to a quadratic function that is critical for pinpointing potential local maxima and minima.

Cubic Function

A cubic function is a type of polynomial function of degree three, typically written in the form f(x) = ax³ + bx² + cx + d with a ? 0. This function is fundamental in algebra and calculus, exhibiting behaviors such as inflection points and potentially multiple turning points due to its odd degree.

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