Problem 61 - Calculus with Concepts in Calculus

Question

Find a function \(f\) such that \(f^{\prime}(0)=0\) and \(f^{\prime \prime}(0)\) does not exist, and such that a. \(f(0)\) is a relative minimum value b. \(f(0)\) is not a relative extreme value

Step-by-Step Solution

Answer

For scenario (a), use \(f(x) = |x^3|\). For scenario (b), use \(f(x) = x^3\).

Step 1: Understanding the Problem

We need to find a function \(f(x)\) such that its first derivative at \(x=0\) is zero, and the second derivative at \(x=0\) does not exist. Additionally, we must create two scenarios: one where \(f(0)\) is a relative minimum and another where \(f(0)\) is not a relative extremum.

Step 2: Creating Functions for Each Scenario

To address scenario (a), consider the function \(f(x) = |x^3|\). This function is continuous, and using derivatives we can show the behavior at \(x=0\). For scenario (b), a suitable function is \(f(x) = x^3\), as it has no extremum at \(x=0\).

Step 3: Checking Derivatives for Scenario (a)

For \(f(x) = |x^3|\), calculate the first derivative. We find that \(f'(x) = 3x^2\cdot \text{sgn}(x)\). At \(x=0\), the derivative \(f'(0) = 0\). The second derivative involves a derivative of a sign function, which is undefined at \(x=0\). Therefore, \(f''(0)\) does not exist.

Step 4: Verifying Minimum for Scenario (a)

Since \(|x^3|\) is always non-negative and attains its lowest value at \(x=0\), \(|x^3|\) is a relative minimum at \(x=0\).

Step 5: Checking Derivatives for Scenario (b)

For \(f(x) = x^3\), calculate the first derivative: \(f'(x) = 3x^2\). At \(x=0\), the derivative \(f'(0) = 0\). The second derivative is \(f''(x) = 6x\), which equals \(0\) at \(x=0\), but it also shows the function doesn't change concavity, confirming no extremum exists.

Step 6: Verifying Non-extremum for Scenario (b)

Since the sign of \(f''(x)\) does not change around \(x=0\), and because the function is cubic (odd degree), \(x=0\) is not a relative extremum for \(x^3\).

Key Concepts

Function Continuity

Continuity is a fundamental property of a function that ensures it behaves predictably without any sudden jumps or breaks. A function is continuous if you can draw its graph without lifting a pen from the paper. In the context of the exercise, continuity plays a crucial role when analyzing functions for extrema and derivatives.

• For scenario (a), we have the function \(f(x) = |x^3|\), which is continuous for all real numbers. The absolute value smoothens out any abrupt changes that might occur at \(x=0\), making the function seamless.
• In scenario (b), \(f(x) = x^3\) is naturally continuous as it is a polynomial, and polynomials are always continuous everywhere on their domain.

Continuity ensures that these functions are well-behaved and allows us to apply derivative tests effectively. Understanding how continuity interacts with other function properties helps in predicting and confirming the behavior of the function at key points.

Derivative Calculation

Derivatives are the backbone of calculus, allowing us to understand how a function changes at any given point. The first derivative \(f'(x)\) gives us the slope or the rate of change of the function, and is responsible for determining where relative extrema might occur. If \(f'(x) = 0\), the function has a stationary point at \(x\).

• For \(f(x) = |x^3|\), the first derivative is \(f'(x) = 3x^2 \cdot \text{sgn}(x)\) with \(f'(0) = 0\). \(\text{sgn}(x)\) is the sign of \(x\), which complicates the second derivative as \(f''(0)\) becomes undefined due to the behavior at \(x=0\).
• For \(f(x) = x^3\), the first derivative is \(f'(x) = 3x^2\), showing that \(f'(0) = 0\) as well. However, its second derivative, \(f''(x) = 6x\), is zero at \(x=0\), illustrating no change in concavity and confirming no relative extremum.

Calculating derivatives gives valuable insight into the function's behavior around critical points and helps in understanding where minimas or maximas might be located.

Relative Extrema

Relative extrema are critical points where a function reaches a local minimum or maximum within a particular interval. At these points, the derivative is zero, indicating no immediate change in direction.

• In scenario (a), \(f(x) = |x^3|\) demonstrates a relative minimum at \(x=0\). This is because the function is non-negative, and \(|x^3|\) reaches its lowest value, zero, at the origin.
• For scenario (b), \(f(x) = x^3\) does not yield a relative extremum at \(x=0\). The function's second derivative analysis reveals no concavity shifts, confirming that \(x=0\) is simply a stationary inflection point, not an extremum.

By understanding how derivatives influence extrema, we can identify critical points where the function might stabilize or change direction, giving a clear picture of the function's local behavior.