Problem 34 - Algebra 2

Question

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=x^{3}+x^{2}-7 x-3 $$

Step-by-Step Solution

Answer

The estimated \(x\)-coordinates of maxima and minima are \(x \approx 1.23\) and \(x \approx -1.90\).

Step 1: Understand the Function

The function given is a cubic polynomial: \(f(x) = x^3 + x^2 - 7x - 3\). Our task is to find the points where this function reaches its maximum and minimum values. This means finding the \(x\)-coordinates where the slope of the tangent to the curve is zero (critical points).

Step 2: Find the Derivative

Differentiate the function to find \(f'(x)\). The derivative \(f'(x)\) represents the slope of the tangent line. So, \(f'(x) = 3x^2 + 2x - 7\).

Step 3: Solve the Derivative Equation

To find the critical points, set the derivative equal to zero: \(3x^2 + 2x - 7 = 0\). Solving this quadratic equation will provide the \(x\)-coordinates where potential maxima or minima occur.

Step 4: Use the Quadratic Formula

Solve the equation \(3x^2 + 2x - 7 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 2\), and \(c = -7\). Compute the solutions to find the \(x\)-coordinates of the critical points.

Step 5: Calculate the Discriminant

The discriminant \(b^2 - 4ac\) is \(2^2 - 4 \times 3 \times (-7)\). Evaluate this to determine if the equation has real roots.

Step 6: Solve for Critical Points

Solve \(x = \frac{-2 \pm \sqrt{88}}{6}\). This simplifies to \(x = \frac{-2 + 9.38}{6}\) and \(x = \frac{-2 - 9.38}{6}\). Thus, the critical points are approximately \(x = 1.23\) and \(x = -1.90\).

Step 7: Use a Graphing Calculator

Input the function \(f(x) = x^3 + x^2 - 7x - 3\) into the graphing calculator. Use the calculator to confirm that these \(x\)-coordinates are where the maxima and minima occur, rounding to the nearest hundredth.

Key Concepts

Cubic Polynomial

A cubic polynomial is an expression that involves terms up to the third degree. This generally has the form \[f(x) = ax^3 + bx^2 + cx + d\]where \(a\), \(b\), \(c\), and \(d\) are constants and \(a eq 0\). The cubic polynomial in our problem is \[f(x) = x^3 + x^2 - 7x - 3\].These types of functions are interesting because they can have up to three real roots and display a wide range of behaviors, like having up to two turning points that could be a maximum or minimum.

• Cubic polynomials can have an 'S-shaped' curve
• They may intercept the x-axis at one, two, or three points based on their roots
• They are not symmetric around any axis unless modified

Understanding the behavior of cubic polynomials is crucial for predicting and estimating the position of these turning points, called critical points.

Critical Points

Critical points of a function occur where the derivative is zero or undefined, indicating a potential maximum, minimum, or saddle point. For the cubic polynomial given \[f(x) = x^3 + x^2 - 7x - 3\],we want to find the critical points where the slope of the tangent line equals zero. This involves taking the derivative of the function. To determine the critical points analytically, you follow these steps:

• Find the derivative of the function, \(f'(x)\).
• Set \(f'(x) = 0\) to find points where the slope is zero.
• Solve the resulting equation for \(x\) to get the critical points.

Finding these critical points helps us understand where the function changes direction - a key step before confirming whether these points are maxima, minima, or neither using further tests or a graphing calculator.

Quadratic Formula

The quadratic formula is a tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides the solution:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In the context of our problem, once we have derived the derivative \(f'(x) = 3x^2 + 2x - 7\), we set this equal to zero to find:\[3x^2 + 2x - 7 = 0\].Here:

• \(a = 3\), \(b = 2\), \(c = -7\)
• The discriminant \(b^2 - 4ac\) determines the nature of the roots
• A positive discriminant means two real roots exist, which is true in our exercise

Applying the quadratic formula, we obtain the critical points \(x \approx 1.23\) and \(x \approx -1.90\), which pinpoint where the extrema might be located.

Derivative

The derivative of a function reflects how it changes at any given point. It's the mathematical tool that tells us the slope of the tangent line and is a fundamental concept when studying rate of changes or slopes.For the function \(f(x) = x^3 + x^2 - 7x - 3\), the derivative is calculated as follows:\[ f'(x) = 3x^2 + 2x - 7 \]This derivative:

• Helps determine the critical points by solving \(f'(x) = 0\)
• Is essential in understanding the function’s concavity and intervals of increase or decrease
• Guides us in utilizing a graphing calculator for visual analysis

By knowing where the derivative is zero, we find the x-values at which the function has potential turning points, crucial before graphing the function to visualize these points.

Graphing Functions

Graphing functions is a visual method that allows us to explore the shape and behavior of mathematical functions. A graphing calculator is particularly useful here, as it can efficiently plot the function and verify estimates of critical points.For our specific function, \(f(x) = x^3 + x^2 - 7x - 3\), the steps to use a graphing calculator include:

• Enter the function in the calculator's graphing mode
• Use the features to locate maxima and minima around the critical points found analytically
• Observe the graph to confirm points of interest (rounded: \(x \approx 1.23\) and \(x \approx -1.90\))

Using a graphing calculator simplifies finding where the function's curve peaks and troughs, aiding our understanding of the function’s overall structure.