Problem 30 - The Calculus with Analytic Geometry

Question

Find formulas for \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) and state the domains of \(f^{\prime}\) and \(f^{\prime \prime}\). $$ f(x)=|x|^{3} $$

Step-by-Step Solution

Answer

The first derivative is \( f'(x) = \begin{cases} 3x^2 & x \geq 0 \ -3x^2 & x < 0 \end{cases} \) with domain \( (-\infty, \infty) \). The second derivative is \( f''(x) = \begin{cases} 6x & x \geq 0 \ -6x & x < 0 \end{cases} \) with domain \( (-\infty, \infty) \).

Step 1 - Understand the Given Function

The given function is \[ f(x) = |x|^3 \].We need to find the first and second derivatives and state their domains.

Step 2 - Split the Absolute Value Function

Rewrite the function without the absolute value sign by considering its definition over different intervals: \[ f(x) = \begin{cases} x^3 & x \geq 0 \ (-x)^3 = -x^3 & x < 0 \end{cases} \]

Step 3 - Differentiate on the Interval \(x \geq 0\)

For \(x \geq 0\), we have \[ f(x) = x^3 \].The first derivative is \[ f'(x) = 3x^2 \].

Step 4 - Differentiate on the Interval \(x < 0\)

For \(x < 0\), we have \[ f(x) = -x^3 \].The first derivative is \[ f'(x) = -3x^2 \].

Step 5 - Combine the First Derivatives

Given the different derivatives in each interval, the combined first derivative function is: \[ f'(x) = \begin{cases} 3x^2 & x \geq 0 \ -3x^2 & x < 0 \end{cases} \]

Step 6 - Identify the Domain of the First Derivative

The first derivative \[ f'(x) = \begin{cases} 3x^2 & x \geq 0 \ -3x^2 & x < 0 \end{cases} \] is defined for all real numbers. Therefore, the domain of \(f'(x)\) is \( (-\infty, \infty) \).

Step 7 - Differentiate Again for the Second Derivative

Differentiate the first derivative function: For \(x \geq 0\), \[ f''(x) = \frac{d}{dx}(3x^2) = 6x \],and for \(x < 0\), \[ f''(x) = \frac{d}{dx}(-3x^2) = -6x \].

Step 8 - Combine the Second Derivatives

Combine the second derivative functions: \[ f''(x) = \begin{cases} 6x & x \geq 0 \ -6x & x < 0 \end{cases} \]

Step 9 - Identify the Domain of the Second Derivative

The second derivative \[ f''(x) = \begin{cases} 6x & x \geq 0 \ -6x & x < 0 \end{cases} \] is defined for all real numbers. Therefore, the domain of \( f''(x) \) is \( (-\infty, \infty) \).

Key Concepts

first derivative

The first derivative of a function, often denoted as \( f'(x) \), represents the rate at which the function's value changes concerning changes in its input value, \(x\). For our function, \( f(x) = |x|^3 \), we need to first differentiate based on the definition of absolute values over different intervals.

When \( x \geq 0 \):

• \( f(x) = x^3 \)
• First derivative: \( f'(x) = 3x^2 \)

When \( x < 0 \):

• \( f(x) = -x^3 \)
• First derivative: \( f'(x) = -3x^2 \)

The combined first derivative function is:
\[ f'(x) = \begin{cases} 3x^2 & x \geq 0 \ -3x^2 & x < 0 \end{cases} \]

This combination ensures our first derivative is piecewise-defined based on the intervals of \( x \).

second derivative

The second derivative, denoted as \( f''(x) \), represents the rate at which the first derivative's value changes with respect to \( x \). In essence, it shows the curvature or concavity of the graph of \( f(x) \).

For our piecewise function \( f'(x) \), we differentiate once more:

When \( x \geq 0 \):

• \( f'(x) = 3x^2 \)
• Second derivative: \( f''(x) = 6x \)

When \( x < 0 \):

• \( f'(x) = -3x^2 \)
• Second derivative: \( f''(x) = -6x \)

The combined second derivative function becomes:
\[ f''(x) = \begin{cases} 6x & x \geq 0 \ -6x & x < 0 \end{cases} \]

This also is piecewise-defined, like the first derivative, and it indicates different behaviors of the function for positive and negative \( x \) values.

absolute value function

The absolute value function describes how to handle real numbers, ensuring they are always non-negative. For any real number \( x \):
\[ |x| = \begin{cases} x & x \geq 0 \ -x & x < 0 \end{cases} \]
In our function, it means:

• For non-negative values (\( x \geq 0 \)), taking the cube of \( x \) directly.
• For negative values (\( x < 0 \)), applying negation transforming \( x \) to \( -x \) and then cubing.

Thus, \( f(x) = |x|^3 \) divides into different expressions based on the sign of \( x \), ensuring both sides of the axis are mirrored cubically but in opposite directions.

domain

The domain of a function describes all possible input values (\(x\)) for which the function is defined. For our initial function, \( f(x) = |x|^3 \), any real number \( x \) can be used, so the domain is: \[ (-\infty, \infty) \]

When considering the derivative functions, we find the first and second derivatives:

• First derivative \( f'(x) \): defined for all real numbers, so domain is \( (-\infty, \infty) \).
• Second derivative \( f''(x) \): also defined for all real numbers, giving a domain of \( (-\infty, \infty) \).

Hence, any value of \( x \) is accepted in both derivations and original function.