Problem 11 - Calculus Early Transcendentals

Question

Locate the critical points and identify which critical points are stationary points. $$ f(x)=\sqrt[3]{x^{2}-25} $$

Step-by-Step Solution

Answer

Critical points are at \( x = 0, 5, -5 \); stationary point is \( x = 0 \).

Step 1: Find the derivative of the function

To find the critical points, we first need the derivative of the function. The function is given as \( f(x) = (x^2 - 25)^{1/3} \). Using the chain rule, the derivative is \( f'(x) = \frac{1}{3}(x^2 - 25)^{-2/3} \cdot 2x \). Simplifying, this becomes \( f'(x) = \frac{2x}{3(x^2 - 25)^{2/3}} \).

Step 2: Find the critical points

Critical points occur where the derivative is zero or undefined. Set \( f'(x) = 0 \), which gives \( \frac{2x}{3(x^2 - 25)^{2/3}} = 0 \). Here, the numerator must be zero, so \( 2x = 0 \), hence \( x = 0 \) is a critical point. Now, find where the derivative is undefined. It is undefined when \( x^2 - 25 = 0 \), which means \( x^2 = 25 \), leading to \( x = 5 \) or \( x = -5 \). So, the critical points are \( x = 0, 5, -5 \).

Step 3: Identify stationary points

A stationary point is when the derivative equals zero. From Step 2, we found \( x = 0 \) makes \( f'(x) = 0 \), so \( x = 0 \) is a stationary point. The other critical points, \( x = 5 \) and \( x = -5 \), are not stationary as the derivative is undefined at these points.

Key Concepts

Critical Points

In calculus, a critical point of a function is where the derivative is either zero or does not exist. These points are important because they can help identify where the function's graph has peaks, valleys, or other significant features. To find the critical points of a function, we first take its derivative and then solve for when this derivative equals zero or is undefined.

For example, consider the function \( f(x) = \sqrt[3]{x^2 - 25} \). To find its critical points, we start by finding the derivative \( f'(x) \) using the chain rule. We then set \( f'(x) = 0 \) and find any \( x \) values that cause the derivative to become zero, these become our critical points. In this case, that's when \( 2x = 0 \), so \( x = 0 \) is a critical point. We also explore where the derivative might be undefined, such as when \( x^2 - 25 = 0 \), giving additional points \( x = 5 \) and \( x = -5 \).

Thus, critical points for \( f(x) = \sqrt[3]{x^2 - 25} \) include \( x = 0, 5, \) and \(-5\).

Derivative

The derivative of a function is a foundational concept in calculus that represents the rate of change or the slope of the function at any given point. To calculate the derivative, especially for more complex functions like \( f(x) = \sqrt[3]{x^2 - 25} \), we often use techniques like the chain rule.

The chain rule is useful for differentiating compositions of functions. In our example, we express the given function as \( (x^2 - 25)^{1/3} \). To apply the chain rule, we differentiate the outer function, in this case, a cubic root, and multiply it by the derivative of the inner function \( x^2 - 25 \). Thus, \( f'(x) = \frac{1}{3}(x^2 - 25)^{-2/3} \cdot 2x \).

After simplifying, the derivative becomes \( f'(x) = \frac{2x}{3(x^2 - 25)^{2/3}} \). This derivative helps identify critical points since it shows where the function's slope is zero or where the derivative is undefined due to division by zero.

Stationary Points

Stationary points are a specific type of critical point where the derivative of the function equals zero. These points are particularly interesting because they can indicate the location of local maxima, minima, or points of inflection in a function. In simpler terms, at stationary points, the function doesn't rise or sink at that exact location; it "rests".

For the function \( f(x) = \sqrt[3]{x^2 - 25} \), when we computed the derivative, we found that one of the solutions was \( x = 0 \), where the derivative is zero. This is what qualifies \( x = 0 \) as a stationary point.

The other critical points \( x = 5 \) and \( x = -5 \) do not turn the derivative to zero but instead make it undefined. Therefore, they aren't classified as stationary points. Remember, understanding where a function's slope turns zero can provide significant insights into the graph's shape and behavior.