Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider \(f(x)=A x^{3}+B x^{2}+C x+D\) with \(A>0\). Show that \(f\) has one local maximum and one local minimum if and only if \(B^{2}-3 A C>0\).

Step-by-Step Solution

Answer

The function has one local maxima and one minima if and only if \(B^2 - 3AC > 0\).

Step 1: Find the Derivative

To find the local maxima and minima of the function \(f(x) = Ax^3 + Bx^2 + Cx + D\), we first calculate its derivative. The derivative \(f'(x)\) is given by: \[ f'(x) = 3Ax^2 + 2Bx + C. \]

Step 2: Calculate Critical Points

To find the critical points, we need to set \(f'(x) = 0\). Thus, we solve the quadratic equation: \[ 3Ax^2 + 2Bx + C = 0. \]Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we substitute \(a = 3A\), \(b = 2B\), and \(c = C\):\[ x = \frac{-2B \pm \sqrt{(2B)^2 - 4 \cdot 3A \cdot C}}{6A} = \frac{-2B \pm \sqrt{4B^2 - 12AC}}{6A}. \]

Step 3: Ensure Two Distinct Real Roots

For \(f\) to have distinct local maximum and minimum, the quadratic equation must have two distinct real roots. This requires the discriminant of the quadratic, \(4B^2 - 12AC\), to be positive. Simplifying, this condition turns into:\[ B^2 - 3AC > 0. \]

Step 4: Conclusion

If \(B^2 - 3AC > 0\), the derived quadratic equation has two distinct solutions, signifying one local maximum and one local minimum exist for \(f\). Conversely, if \(B^2 - 3AC \leq 0\), either no real solutions or a repeated root exists, suggesting no distinct local extremum pair.

Key Concepts

Derivative of Polynomial Functions

Understanding the derivative of polynomial functions is critical to analyzing their behavior and identifying important characteristics such as local maxima and minima. The derivative of a polynomial function provides the rate at which the function's value changes with respect to changes in the input variable. For a polynomial of the form \(f(x) = Ax^3 + Bx^2 + Cx + D\), the derivative is calculated by applying the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\).
Thus, the derivative of our given polynomial is:

• For the term \(Ax^3\), the derivative is \(3Ax^2\).
• The derivative of \(Bx^2\) is \(2Bx\).
• The derivative of \(Cx\) is simply \(C\).
• The constant term \(D\) disappears, as its derivative is zero.

The result combines all these derivatives into a new polynomial function: \(f'(x) = 3Ax^2 + 2Bx + C\). This derivative function helps us find the critical points that determine where the graph of \(f(x)\) has potential maxima, minima, or points of inflection.

Quadratic Equation Discriminant

The discriminant of a quadratic equation plays a pivotal role in determining the nature of the equation's roots. For quadratic equations of the standard form \(ax^2 + bx + c = 0\), the discriminant is defined as \(b^2 - 4ac\).
In the context of our polynomial's derivative \(f'(x) = 3Ax^2 + 2Bx + C\), solving the quadratic equation allows us to find critical points. Here, the discriminant is calculated from the coefficients:

• \(a = 3A\)
• \(b = 2B\)
• \(c = C\)

Substituting these into the discriminant formula, we have \((2B)^2 - 4 \cdot 3A \cdot C\), which simplifies to \(4B^2 - 12AC\).
This result must be greater than zero, \(B^2 - 3AC > 0\), for the quadratic to have two distinct real roots. These roots correspond to the points of local maximum and minimum of our original cubic function.

Local Maximum and Minimum

Local maxima and minima are points on a graph where a function reaches a peak or a trough respectively. They are critical in understanding the behavior of functions across different intervals. For a function to have a local maximum and minimum, there must be a change in the function's increasing and decreasing intervals, which is identified by the critical points derived from its derivative.
In our case, the critical points are the solutions to the equation \(f'(x) = 0\), which we solved using the quadratic formula. The discriminant \(B^2 - 3AC > 0\) ensures that we have two distinct solutions, meaning that the function changes behavior at these points:

• One critical point will correspond to a local maximum where the function value peaks within a certain interval.
• The other will indicate a local minimum, where the function dips to a local low point.

These features are essential for graphically understanding how the cubic function behaves, and they illustrate how calculus can predict and explain the function's turning points.