Problem 49 - University Calculus: Early Transcendentals

Question

Determine all critical points for each function. $$y=x^{2}+\frac{2}{x}$$

Step-by-Step Solution

Answer

Critical points are at \(x = 1\) and \(x = 0\).

Step 1: Finding the Derivative

To find the critical points, we start by finding the first derivative of the function. The function given is \(y = x^2 + \frac{2}{x}\). Using the rules for derivatives, we have \[\frac{dy}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}\left(\frac{2}{x}\right)\].The derivative of \(x^2\) is \(2x\), and the derivative of \(\frac{2}{x}\) is \(-\frac{2}{x^2}\). Therefore, the first derivative is:\[\frac{dy}{dx} = 2x - \frac{2}{x^2}\].

Step 2: Setting the Derivative Equal to Zero

The critical points occur where the derivative is equal to zero or undefined. First, we solve \[2x - \frac{2}{x^2} = 0\].To eliminate the fraction, multiply each term by \(x^2\) to get \[2x^3 - 2 = 0\].

Step 3: Solving for Critical Points

From the equation in Step 2, \(2x^3 - 2 = 0\), add 2 to both sides to obtain \[2x^3 = 2\].Divide both sides by 2:\[x^3 = 1\].Take the cube root of both sides: \[x = 1\].

Step 4: Identifying Undefined Points

The derivative \(\frac{dy}{dx} = 2x - \frac{2}{x^2}\) is undefined when the denominator of any term is zero. In this case, \(\frac{2}{x^2}\) is undefined when \(x = 0\). Therefore, \(x = 0\) is another point to consider.

Step 5: Listing All Critical Points

The critical points are where the derivative is zero or undefined. We found that \(x = 1\) makes the derivative zero and \(x = 0\) makes it undefined. The critical points for the function \(y = x^2 + \frac{2}{x}\) are \(x = 1\) and \(x = 0\).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus. It helps us determine the rate at which a function changes at any given point. For a function like \(y = x^2 + \frac{2}{x}\), finding its derivative requires an application of basic rules of differentiation.
When calculating the derivative:\( \frac{dy}{dx} \), we differentiate each term separately:

• The derivative of \(x^2\) is \(2x\). This follows from the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\).
• The derivative of \(\frac{2}{x}\) can be rewritten using a negative exponent: \(2x^{-1}\). According to the power rule, this becomes \(-2x^{-2}\) or \(-\frac{2}{x^2}\).

Thus, the derivative of the entire function is \[ \frac{dy}{dx} = 2x - \frac{2}{x^2} \].
Understanding derivatives is crucial because it helps us analyze the behavior of the function, including identifying critical points and potential changes in the function's increasing or decreasing nature.

Undefined Points

In calculus, certain values of \(x\) can lead to a derivative being undefined. Undefined points occur when a mathematical operation leads to indeterminacy, such as division by zero. In analyzing the derivative \(\frac{dy}{dx} = 2x - \frac{2}{x^2}\), observe the term \(\frac{2}{x^2}\). This term is undefined when \(x = 0\) because division by zero is not possible.
Identifying undefined points is essential because they can indicate discontinuities or abrupt changes in the behavior of the function. While \(x=0\) makes the second term undefined, it's a point of interest and must be considered alongside traditional critical points where the derivative is zero.
Analyzing undefined points often involves checking the function's graph or behavior approaching that point from both directions. This helps ensure complete understanding of how the function behaves as \(x\) approaches the undefined point.

Function Analysis

Function analysis in calculus involves examining the overall behavior of a given function. This includes finding derivatives, identifying critical points, and understanding where a function is increasing or decreasing.
For the function \(y = x^2 + \frac{2}{x}\), we determine critical points from the derivative calculation. These occur when:

• The derivative \(\frac{dy}{dx} = 2x - \frac{2}{x^2} \) is equal to zero. Solving \(2x - \frac{2}{x^2} = 0\) reveals that \(x = 1\) is a critical point where the function's slope is zero.
• The derivative is undefined, such as at \(x = 0\).

Critical points like \(x = 1\) and undefined points like \(x = 0\) inform us about potential maxima, minima, or inflection points in the function.
In practice, understanding these points allows us to sketch the graph of the function more accurately and predict its behavior under various conditions. By studying critical points, we gain insight into optimization problems and changes in direction in the function's graph.