Question

Cubic Polynomial Functions Let $$f(x)=a x^{3}+b x^{2}+c x+d, a \neq 0$$ (a) Show that f has either 0 or 2 local extrema. (b) Give an example of each possibility in part (a).

Short answer

For a cubic polynomial function \(f(x) = a x^{3}+b x^{2}+c x+d\), it is proven that function f(x) can have either 0 or 2 local extrema. Two examples supporting the statement include \(f(x) = x^3 - 3x + 2\) having 2 local extremum and \(f(x) = x^3\) having no local extremum.

Step-by-step solution

Calculating the Derivative of the Function

Calculate the first derivative of the function \(f(x) = a x^{3}+b x^{2}+c x+d\). As per the power rule, the derivative of \(f(x)\) is \(f'(x) = 3a x^2 + 2b x+ c\).

Checking for Local Extrema

To determine the loci of local extremum, set the first derivative equal to zero and solve for x. These points are where the function has local extremum, provided the second derivative is not zero. So, solving \(f'(x) = 3a x^2 + 2b x+ c = 0\) would give us the x-values of the extremum.

Determining the Number of Extrema

To find the number of solutions to this, determine the discriminant of the equation \(3a x^2 + 2b x+ c = 0\). This is given by the formula \(D = b^2 - 4ac\). If D > 0, there are 2 solutions (2 local extremum). If D < 0, there are no real solutions (0 local extremum). We thus prove that the cubic function has either 0 or 2 local extrema.

Providing examples

For Part (b), one could choose \(f(x) = x^3 - 3x + 2\) as an example of a cubic function with two local extrema, by calculating its derivative \(f'(x) = 3x^2 - 3\) and finding the roots as -1 and 1. For a function with no local extremum, one could choose \(f(x) = x^3\), for which \(f'(x) = 3x^2\) has no real roots, hence it has no local extrema.

Key concepts

Calculating Derivatives

Derivatives form the core of calculus and provide a way to describe the rate at which functions change. Calculating a derivative means finding a new function that gives the slope of the original function at any given point. For polynomial functions, such as cubic functions, the process is streamlined by the power rule.

The power rule states that if you have a function like \( f(x) = x^n \), its derivative, denoted as \( f'(x) \), would be \( nx^{n-1} \). Applying this to the cubic polynomial \( f(x) = ax^3 + bx^2 + cx + d \), we differentiate each term separately to get \( f'(x) = 3ax^2 + 2bx + c \). This derivative function is what we use to find slopes and rates of change at any point on the curve of the cubic polynomial.

Local Extrema of Functions

Local extrema refer to the peaks and valleys in the graph of a function - points where the function reaches a local maximum or minimum. To find these points for a cubic polynomial, we set its first derivative equal to zero because local extrema occur where the slope of the tangent to the function's graph is zero.

By solving the equation \( f'(x) = 3ax^2 + 2bx + c = 0 \), we're finding the x-values where the function's rate of change switches direction, which indicates a local max or min. However, we must ensure that these points are indeed extrema by checking the second derivative or by using the first derivative test.

Discriminant of Quadratic Equation

When analyzing the local extrema of cubic functions, we often encounter quadratic equations. The discriminant is a component of the quadratic formula that provides critical information about the nature of the solutions to a quadratic equation.The discriminant, denoted as \( D \), is found using the formula \( D = b^2 - 4ac \). If \( D > 0 \), there are two distinct real solutions. If \( D = 0 \), there is exactly one real solution (a repeated root). And if \( D < 0 \), there are no real solutions, implying the graph of the function does not cross the x-axis. In the context of finding local extrema for cubic functions, the sign of the discriminant tells us how many extrema the function has.

Power Rule in Calculus

The power rule is a basic, yet powerful, technique in calculus used to find derivatives of functions where the variable has an exponent. It simplifies the process of differentiation for any term of the form \( ax^n \). According to the power rule, the derivative of this term is \( anx^{n-1} \).

Thus, for a cubic term like \( ax^3 \), the derivative would be \( 3ax^2 \). The power rule not only makes it easy to calculate derivatives for polynomials but also paves the way for studying the behavior of functions, such as finding their rates of change and locating local extrema.

Graphing Cubic Functions

Graphing cubic functions, which are functions of the form \( f(x) = ax^3 + bx^2 + cx + d \), involves understanding their shape and behavior. These graphs are known for their distinct 'S-curve' shape. The coefficients of the equation influence how the graph looks - whether it opens up or down, the location and number of turns, and the steepness of the curve.

To sketch the graph of a cubic function, it helps to calculate the function's derivatives to identify local extrema, points of inflection, and the overall slope behavior at different intervals. By plotting these features, along with any x-intercepts and the y-intercept, you can accurately draw the curve of a cubic polynomial.