Problem 49 - Calculus

Question

Explain what is wrong with the statement. The function \(y=2 x\) is increasing, and its graph is concave up.

Step-by-Step Solution

Answer

The statement is incorrect because linear functions like \(y=2x\) are straight lines with no concavity.

Step 1: Understand the Function Type

The function given is a linear function of the form \(y = mx + b\), where \(m = 2\) and \(b = 0\). Linear functions are neither concave up nor concave down; they are straight lines. In this case, "concave up" is not applicable to the function since it has no curvature at all.

Step 2: Identify the Slope

In the function \(y = 2x\), the slope \(m\) is equal to 2, which is positive. This indicates that the function is increasing, which means for any \(x_1 < x_2\), \(f(x_1) < f(x_2)\). The statement about the function increasing is correct.

Step 3: Investigate Concavity

Concavity refers to the direction of curvature of a graph. For a function to be concave up, its second derivative must be positive. Since \(y = 2x\) is a linear function, its second derivative is zero, indicating no concavity. Thus, the statement that the graph is concave up is incorrect.

Key Concepts

Concavity in Linear Functions

In mathematics, concavity describes the curvature of a graph. For a function to be "concave up," it means the graph looks like a U-shape, bending upwards. Similarly, being "concave down" means the graph resembles an upside-down U, bending downwards.
Linear functions, like the function given in the form of \(y = 2x\), are unique because they do not curve.
They are straight lines. Therefore, talking about concavity in linear functions is misleading. Concavity is determined by the second derivative of a function. In linear functions, the second derivative is always zero, which means no concavity is present.

• Linear functions: no curvature.
• Second derivative is zero, indicating no concavity.

Remember, concavity is relevant for functions that curve, such as quadratic functions or exponential functions, but not for straight lines!

Understanding Slope

The slope of a line is a measure of its steepness and direction. In the linear equation \(y = 2x\), the slope is represented by the coefficient of \(x\), which is 2.
This slope is positive, meaning the line rises as it moves from left to right. This is why the function \(y = 2x\) is considered an "increasing" function.
Whenever you see a positive slope:

• The line goes upwards.
• The function values increase as \(x\) increases.

A positive slope shows an upward trend, and thus, they indicate that the function is increasing. In contrast, a negative slope would mean the line goes downwards, denoting a "decreasing" function. Remember, the slope is crucial when analyzing directions and trends in linear functions.

The Role of the First Derivative

The first derivative of a function gives us information about its rate of change, or its slope at any point on the graph. For linear functions like \(y = 2x\), this derivative is constant.
Taking the derivative of \(y = 2x\), we find \(\frac{d}{dx}(2x) = 2\). This means that the slope is uniformly 2 everywhere on the graph.

• Describes the slope (rate of change).
• For \(y = 2x\), the derivative is 2 (constant).

The first derivative helps us understand how the output of a function changes relative to its input. Since linear functions have a constant rate of change, their first derivative remains unchanged across all points in the domain.

Exploring the Second Derivative

The second derivative of a function provides insight into its concavity. It tells us whether the function is bending upwards or downwards. For linear functions such as \(y = 2x\), this consecutive differentiation yields zero.
When we differentiate the first derivative (which is 2 in this case), we get zero: \(\frac{d^2}{dx^2}(2x) = 0\).

• Indicates concavity or lack thereof.
• For linear equations, second derivative is zero, meaning no curvature.

Because linear functions are perfectly straight with no bends, the second derivative is consistently zero, implying that they are neither concave up nor concave down. This clarifies why concavity doesn't apply to linear functions.