Problem 26 - Essential Calculus Early Transcendentals

Question

\(23-36=\) Find the critical numbers of the function. $$f(x)=2 x^{3}+x^{2}+2 x$$

Step-by-Step Solution

Answer

The function has no critical numbers as there are no real roots for the derivative equation.

Step 1: Differentiate the Function

To find the critical numbers of the function \(f(x) = 2x^3 + x^2 + 2x\), first compute the derivative \(f'(x)\). Applying the power rule, we get: \[f'(x) = \frac{d}{dx}(2x^3) + \frac{d}{dx}(x^2) + \frac{d}{dx}(2x) = 6x^2 + 2x + 2.\]

Step 2: Set Derivative Equal to Zero

The critical numbers occur where the derivative \(f'(x)\) is equal to zero or is undefined. Since \(f'(x) = 6x^2 + 2x + 2\) is a polynomial, it's defined everywhere, so set \(f'(x) = 0\): \[6x^2 + 2x + 2 = 0.\]

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \(6x^2 + 2x + 2 = 0\) for \(x\) using the quadratic formula. The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where \(a = 6\), \(b = 2\), and \(c = 2\). First, compute the discriminant: \[b^2 - 4ac = 2^2 - 4 \times 6 \times 2 = 4 - 48 = -44.\]

Step 4: Determine the Roots Based on the Discriminant

Since the discriminant \(-44\) is negative, the quadratic equation \(6x^2 + 2x + 2 = 0\) has no real solutions. This means there are no critical numbers as there are no real values of \(x\) where the derivative is zero.

Key Concepts

Derivative

The derivative of a function provides us with a powerful tool to understand the behavior of the function. In essence, the derivative represents the rate at which the function's value changes as the input changes. In more intuitive terms, it tells us the "slope" of the function at any given point.

To calculate the derivative, we often use rules like the power rule, which is essential for functions that involve polynomials. For example, to differentiate a term like \(x^n\), you simply bring down the exponent \(n\) as a coefficient and subtract one from the exponent, resulting in \(nx^{n-1}\).

In this exercise, finding the derivative \(f'(x) = 6x^2 + 2x + 2\) from the original function \(f(x) = 2x^3 + x^2 + 2x\) is an application of this rule. Recognizing when and how to differentiate a function is crucial for identifying critical numbers, where the function's increasing or decreasing trends change.

Quadratic Formula

The quadratic formula is a staple method in algebra used for solving quadratic equations. These are equations that can be written in the standard form \(ax^2 + bx + c = 0\). The magical property of the quadratic formula is that it gives you the exact solutions—or roots—of the quadratic equation.

The formula itself is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

By plugging in the constants \(a\), \(b\), and \(c\) from the quadratic equation into the formula, you can solve for \(x\).

It's often used because it works regardless of whether you can factor the equation easily or not, making it extraordinarily versatile.

Discriminant

The discriminant is a component of the quadratic formula, notably found under the square root: \(b^2 - 4ac\). This single term can tell you a lot about the nature of the solutions to your quadratic equation.

The discriminant has three key outcomes:

• If it is positive, the equation has two real and distinct solutions.
• If it is zero, there is exactly one real solution (or a repeated root).
• If it is negative, as in our example \(b^2 - 4ac = -44\), the quadratic equation has no real solutions, meaning the roots are complex or imaginary.

Understanding the discriminant is crucial when assessing how a quadratic equation behaves.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of \(x\), each multiplied by a coefficient. These functions are fundamental in mathematics due to their simplicity and versatility. Their general form is \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\).

Polynomial functions can have various degrees denoted by the highest power of \(x\) that appears with a non-zero coefficient. For example, the function \(f(x) = 2x^3 + x^2 + 2x\) is a cubic polynomial (since the highest power is 3).

Such functions are continuous and smooth, which is why they are often used in modeling natural phenomena. They can be analyzed in different ways, such as by finding their derivatives, to understand how they change and identify features like critical numbers, where the function's behavior might change.