Problem 38 - Calculus: Early Transcendentals

Question

Describe the concavity of the functions below. $$ y=x^{4}-2 x^{2}+3 $$

Step-by-Step Solution

Answer

The function is concave up for \( x < -\sqrt{\frac{1}{3}} \) and \( x > \sqrt{\frac{1}{3}} \), and concave down for \( -\sqrt{\frac{1}{3}} < x < \sqrt{\frac{1}{3}} \).

Step 1: Find the second derivative

To determine the concavity of the function, we need to find the second derivative of the function. First, find the first derivative of the given function: \[ y = x^{4} - 2x^{2} + 3 \] The first derivative is: \[ y' = 4x^{3} - 4x \]Next, find the second derivative by differentiating the first derivative:\[ y'' = (4x^{3} - 4x)' = 12x^{2} - 4 \]

Step 2: Analyze the second derivative

The sign of the second derivative determines the concavity. If the second derivative is positive, the function is concave up; if it is negative, the function is concave down. Now set the second derivative equal to zero to find critical points:\[ 12x^{2} - 4 = 0 \]Solving for \( x \) gives:\[ 12x^{2} = 4 \]\[ x^{2} = \frac{1}{3} \]\[ x = \pm \sqrt{\frac{1}{3}} \]

Step 3: Test intervals for concavity

With critical points at \( x = \pm \sqrt{\frac{1}{3}} \), evaluate intervals around these points to determine the concavity.1. Choose a point in the interval \( x < -\sqrt{\frac{1}{3}} \), such as \( x = -1 \): \[ y''(-1) = 12(-1)^{2} - 4 = 8 \] (positive, concave up)2. Choose a point in the interval \( -\sqrt{\frac{1}{3}} < x < \sqrt{\frac{1}{3}} \), such as \( x = 0 \): \[ y''(0) = 12(0)^{2} - 4 = -4 \] (negative, concave down)3. Choose a point in the interval \( x > \sqrt{\frac{1}{3}} \), such as \( x = 1 \): \[ y''(1) = 12(1)^{2} - 4 = 8 \] (positive, concave up)Thus, the function is concave up for \( x < -\sqrt{\frac{1}{3}} \) and \( x > \sqrt{\frac{1}{3}} \), and concave down for \( -\sqrt{\frac{1}{3}} < x < \sqrt{\frac{1}{3}} \).

Key Concepts

Concavity

In calculus, concavity refers to the direction in which a curve bends.
When analyzing a function, determining concavity helps us understand the shape and behavior of the graph.

Simply put, a function is said to be concave up if its graph makes an upward bowl shape. If it makes a downward bowl shape, the function is concave down.
To find where these changes occur, we look at the second derivative of the function.

• If the second derivative is positive ( y'' > 0 ), it means the function is concave up in that interval.
• If the second derivative is negative ( y'' < 0 ), the function is concave down.

This information can be practical for graphing functions, as it tells you where the graph will curve upwards or downwards.
Identification of these patterns is beneficial in optimizing or analyzing changes in business and science-related problems.

Second Derivative

The second derivative is simply the derivative of the derivative of a function.
It provides information about the curvature or concavity of the original function.

When you first take the derivative, you find the slope of the tangent line at any point on the function.
The second derivative goes a step further to detail how this slope is changing.

Mathematically, given a function y = f(x) , the first derivative is y' = f'(x) . The second derivative is then y'' = f''(x) .

• The second derivative can reveal whether the slope is increasing or decreasing.
• It plays a crucial role in determining points of inflection, where the function changes concavity.

Understanding second derivatives is vital for thorough graph analysis and also in physics, describing acceleration when dealing with motion.

Differentiation

Differentiation is the process of finding a derivative, which is the measure of how a function changes as its input changes.
In calculus, the derivative is a foundational concept, crucial for various analyses including rates of change, optimization, and curve sketching.

Given a function y = f(x) , its derivative y' = f'(x) gives the rate at which y changes with respect to x .
For example:

• In physics, derivatives can describe velocity—how fast position changes with time.
• In economics, they can model marginal cost—how cost changes with product quantity.

When extending differentiation to find higher-order derivatives, such as the second derivative, you can uncover deeper properties of the function's graph, including concavity and inflection points.

Critical Points

Critical points on a graph are where the derivative of a function equals zero or is undefined.
These points are significant because they often indicate locations of local maxima, minima, or points of inflection.

To locate critical points, simply take the derivative of the function and solve for when it equals zero or is undefined.

• These are points where the function's slope changes direction.
• After finding critical points, further examination with second derivatives can clarify the nature of these points: whether they signify a peak, a trough, or a change in concavity.

Recognizing and analyzing critical points is an essential skill in calculus, aiding in optimization problems and comprehensive graph analysis.