Problem 68 - Calculus with Concepts in Calculus

Question

Let \(f(x)=x^{3}+a x^{2}+b x+c\), where \(a, b\), and \(c\) are constants. a. Show that \(f\) is increasing on \((-\infty, \infty)\) if \(a^{2} \leq 3 b\). b. Assume that \(a^{2}>3 b\). Show that \(f\) is increasing on $$ \left(-\infty,\left(-a-\sqrt{a^{2}-3 b}\right) / 3\right] $$ and on $$ \left[\left(-a+\sqrt{a^{2}-3 b}\right) / 3, \infty\right) $$ and is decreasing on \(\left[\left(-a-\sqrt{a^{2}-3 b}\right) / 3,\left(-a+\sqrt{a^{2}-3 b}\right) / 3\right]\)

Step-by-Step Solution

Answer

a) Increasing on \((- fty, fty)\) if \(a^2 \leq 3b\). b) Increasing on \((- fty, x_1)\) and \((x_2, fty)\), decreasing on \((x_1, x_2)\) if \(a^2 > 3b\).

Step 1: Find the derivative of f(x)

To determine the intervals where the function is increasing or decreasing, first find the derivative of the function. For the given function \(f(x) = x^3 + ax^2 + bx + c\), the derivative is \(f'(x) = 3x^2 + 2ax + b\).

Step 2: Analyze f'(x) for critical points

Critical points occur where \(f'(x) = 0\). Solve the equation \(3x^2 + 2ax + b = 0\) using the quadratic formula to find potential critical points: \[x = \frac{-2a \pm \sqrt{(2a)^2 - 4 \cdot 3 \cdot b}}{2 \cdot 3}\]. This simplifies to \[x = \frac{-2a \pm \sqrt{4a^2 - 12b}}{6} = \frac{-a \pm \sqrt{a^2 - 3b}}{3}\].

Step 3: Analyze where f'(x) ≥ 0

For part a, if \(a^2 \leq 3b\), the discriminant \((a^2 - 3b)\) is less than or equal to zero, meaning there are no real roots for \(f'(x) = 0\) and thus \(f'(x) > 0\) for all \(x\). Therefore, \(f(x)\) is increasing on \((-fty, \infty)\) because the parabola defined by \(3x^2 + 2ax + b\) opens upwards and does not intersect the x-axis.

Step 4: Analyze where f'(x) changes sign

For part b, if \(a^2 > 3b\), then \(f'(x)\) has two roots given by \(x_1 = \frac{-a - \sqrt{a^2 - 3b}}{3}\) and \(x_2 = \frac{-a + \sqrt{a^2 - 3b}}{3}\). The parabola \(3x^2 + 2ax + b\) opens upwards. Thus, \(f'(x) < 0\) between \(x_1\) and \(x_2\) and \(f'(x) > 0\) on intervals \((-fty, x_1)\) and \((x_2, fty)\).

Step 5: Conclusion

From the previous analysis, you conclude: - If \(a^2 \leq 3b\), \(f\) is increasing on \((-fty, fty)\).- If \(a^2 > 3b\), \(f\) is increasing on \((-fty, x_1)\) and \((x_2, fty)\), and decreasing on \((x_1, x_2)\).

Key Concepts

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is essential in calculus. We determine whether a function is increasing or decreasing by looking at its derivative. If the derivative is positive over an interval, the function is increasing in that interval. If the derivative is negative, the function is decreasing. Consider the function

• Find the derivative.
• Analyze the positivity or negativity of the derivative to determine intervals of increase and decrease.

In our example, given a polynomial equation, we determine it is increasing or decreasing based on the sign of its derivative. This understanding helps us predict the behavior of complex functions over different intervals.

Quadratic Formula

The quadratic formula is a critical tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides the roots of a quadratic equation, which are crucial when analyzing equations derived from derivatives. The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]

• The expression under the square root, known as the discriminant \(b^2 - 4ac\), determines the nature of the roots.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• If negative, the roots are complex.

In the derivative analysis of cubic functions, we use the quadratic formula to find critical points, which are values where the rate of change of the function makes significant shifts.

Critical Points

Critical points occur where the function's derivative equals zero or does not exist. These points help us identify where the function shifts from increasing to decreasing or vice versa. To find critical points:

• Set the derivative equal to zero and solve for \(x\).
• Use the results to check intervals around these points to see where the function changes behavior.

For polynomial functions like our cubic example, these points usually involve using the quadratic formula to solve the derivative's equation. Critical points are foundational in understanding the overall behavior of functions, especially in identifying peaks (maximums) and troughs (minimums).

Derivative Analysis

Derivative analysis involves examining the derivative of a function to understand the function's behavior. The first derivative, \(f'(x)\), gives us the slope or rate of change of the function. Positive slopes indicate increasing functions, while negative slopes indicate decreasing functions. Here are some key steps:

• Compute the first derivative of the function.
• Identify intervals where the derivative is positive or negative.
• Determine any critical points where \(f'(x) = 0\).
• Use these insights to describe the function's behavior.

For our example, the derivative \(f'(x) = 3x^2 + 2ax + b\) is analyzed for increasing or decreasing behavior. Through this, you can determine the impact of different parameters, such as \(a\) and \(b\), on the function's increasing and decreasing intervals. By understanding these concepts, you gain a clear picture of the mathematical landscapes functions create.