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Chapter 5: Problem 48

Describe the concavity of the functions below. $$ y=x^{2}+1 / x $$

Short Answer

Expert verified
The function is concave up for all \( x \neq 0 \).

Step by step solution

01

Find the first derivative

To determine concavity, we first need the first derivative of the function. For the function \[ y = x^2 + \frac{1}{x} \]we use the power rule and the derivative of \(1/x\) to find:\[ y' = 2x - \frac{1}{x^2} \]
02

Find the second derivative

Next, we find the second derivative of the function, which tells us about the concavity. Differentiate \( y' = 2x - \frac{1}{x^2} \) with respect to \( x \):\[ y'' = 2 + \frac{2}{x^3} \]
03

Analyze the second derivative

The concavity is determined by the sign of the second derivative \( y'' \). If \( y'' > 0 \), the function is concave up; if \( y'' < 0 \), it is concave down. The expression:\[ y'' = 2 + \frac{2}{x^3} \]has two parts: 2 (a constant positive term) and \( \frac{2}{x^3} \).
04

Determine the concavity

The term \( \frac{2}{x^3} \) is positive when \( x > 0 \) and negative when \( x < 0 \). Therefore:- For \( x > 0 \), \( y'' = 2 + \frac{2}{x^3} > 0 \), so the function is concave up.- For \( x < 0 \), \( y'' = 2 + \frac{2}{x^3} < 2 \) but still positive, implying the function is concave up.Thus, the function is concave up for all \( x eq 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Concavity
Concavity in calculus refers to the curvature of a function's graph. It's essential in understanding how a function behaves.
When a graph bends upwards like a cup, it's described as **concave up**. Conversely, if it bends downwards like a dome, it's **concave down**. This changes at points called points of inflection.
To analyze concavity, we assess the second derivative of a function. A positive second derivative indicates concave-up behavior, while a negative one indicates concave-down.
Second Derivative
The second derivative serves as a tool to determine concavity. It involves taking the derivative of a function twice. First, we derive the original function to find the first derivative. Then, we derive that result again.
Let's take an example: for the function \( y = x^2 + \frac{1}{x} \). The first derivative \( y' = 2x - \frac{1}{x^2} \) reveals the rate of change. Next, we derive again to get \( y'' = 2 + \frac{2}{x^3} \).
This expression tells us how the first derivative's rate of change varies. It plays a crucial role in analyzing where and how function curves change.
First Derivative
The first derivative represents the slope of the tangent line to a function's graph. Essentially, it shows how rapidly or slowly the function changes. We obtain it by deriving the original function once.
For the function \( y = x^2 + \frac{1}{x} \), applying the power rule and derivative rules gives us \( y' = 2x - \frac{1}{x^2} \). A positive \( y' \) value at a point suggests an increasing function, while a negative value indicates a decreasing function there.
Grasping the concept of the first derivative lays the foundation for finding the second derivative and understanding more intricate details of a function's shape.
Power Rule
The power rule is a standard technique in calculus for finding derivatives. It's straightforward and widely applicable. When differentiating a function in the form \( x^n \), the power rule tells you to multiply by the power and decrease the power by one. In formula terms: if \( y = x^n \), then \( y' = nx^{n-1} \).
This rule simplifies differentiation, especially for polynomials. In our example \( y = x^2 + \frac{1}{x} \), applying the power rule to each term allows us to efficiently find the derivative \( y' = 2x - \frac{1}{x^2} \).
Understanding and applying the power rule is essential for mastering calculus derivatives.

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