Problem 12 - Calculus A New Horizon

Question

Locate the critical points, and classify them as stationary points or points of nondifferentiability. (a) \(f(x)=x^{4 / 3}-6 x^{1 / 3}\) (b) \(f(x)=|\sin x|\)

Step-by-Step Solution

Answer

(a) Critical points: \(x = \frac{3}{2}\) (stationary), \(x = 0\) (nondifferentiable). (b) Critical points at \(n\pi\); stationary and nondifferentiable at \(x = 2k\pi\).

Step 1: Find the derivative of (a)

To find the derivative of the function \( f(x) = x^{4/3} - 6x^{1/3} \), apply the power rule:\[f'(x) = \frac{4}{3}x^{1/3} - 6 \cdot \frac{1}{3}x^{-2/3} = \frac{4}{3}x^{1/3} - 2x^{-2/3}\]

Step 2: Solve for critical points of (a)

Set the derivative from Step 1 equal to zero to find the critical points:\[\frac{4}{3}x^{1/3} - 2x^{-2/3} = 0\]Multiply through by \(x^{2/3}\) to clear the fraction:\[4x - 6 = 0\]Solve for \(x\):\[x = \frac{3}{2}\]

Step 3: Check for points of nondifferentiability in (a)

Examine the derivative \(f'(x)\) for any points of nondifferentiability, which occur where it is undefined:Since \(x^{-2/3}\) is undefined at \(x = 0\), \(x = 0\) is a point of nondifferentiability.

Step 4: Determine the derivative for (b)

The function given is \(f(x) = |\sin x|\). The derivative is:\[f'(x) = \begin{cases} \cos x & \text{if } \sin x > 0 \-\cos x & \text{if } \sin x < 0 \0 & \text{if } \sin x = 0 \end{cases}\]

Step 5: Solve for critical points of (b)

Critical points occur where \(f'(x) = 0 \) or where the function changes differentiability due to absolute value. Since\(|\sin x|\) becomes zero at integer multiples of \(\pi\) (i.e., \(x = n\pi\) for integer \(n\)), these are stationary points.

Step 6: Check for points of nondifferentiability in (b)

\(|\sin x|\) is nondifferentiable where the value of \( \sin x \) changes sign, which occurs at \( x = n\pi \). Thus, the points \( x = 2k\pi \) are both stationary and points of nondifferentiability due to the absolute value.

Key Concepts

Derivative

A derivative represents how a function changes as its input changes. Imagine you're driving a car: the derivative is like measuring your speed, showing how fast you're moving at any given moment. For a mathematical function, it shows how the function value changes with respect to changes in the input value.
For example, consider the derivative of the function in part (a) given as:

• Apply the power rule: The derivative of each term inside is found separately. The first term, \( x^{4/3} \), when differentiated using the power rule, becomes \( \frac{4}{3}x^{1/3} \).
• The second term, \(-6x^{1/3}\), becomes \(-2x^{-2/3}\) after applying the power rule.

This derivative tells us how the slope of the function changes at each point. By setting the derivative to zero, we locate critical points, which are potential points of a maximum, minimum, or inflection on the graph.

Nondifferentiability

Nondifferentiability refers to points where a function does not have a derivative. It's like trying to find the speed of a car when its speedometer is broken—you just can't do it! There are common scenarios for nondifferentiability:

• The function has a sharp point or cusp (like an absolute value function).
• The function has a vertical tangent (where the slope goes infinitely steep).
• The function is not defined at that point.

In the exercises:
• For part (a), nondifferentiability occurs at \(x = 0\) due to the \( x^{-2/3} \) term in the derivative—this is where the derivative goes undefined.
• For part (b), the function \( |\sin x| \) becomes nondifferentiable where the sign changes, such as at \( x = n\pi \).

Stationary Points

Stationary points are where the derivative of a function equals zero. Think of it as finding places where a roller coaster momentarily slows to a stop as it changes direction. At these points, the function could be at a peak, trough, or could level out momentarily.
To identify stationary points in the exercises:

• In part (a), setting the derivative equal to zero shows that \( x = \frac{3}{2} \) is a stationary point.
• For part (b), \( f'(x) = 0 \) at integer multiples of \( \pi \), indicating points like \( x = n\pi \) as stationary.

These points are crucial for understanding the behavior of functions and identifying where it might "pause" before changing direction or behavior. Not every stationary point is a maximum or minimum—further testing is needed to classify them.