Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(g(x)\) be a function that has two derivatives and satisfies the following properties: (a) \(g(1)=1\); (b) \(g^{\prime}(x)>0\) for all \(x \neq 1\); (c) \(g\) is concave down for all \(x<1\) and concave up for all \(x>1\) (d) \(f(x)=g\left(x^{4}\right)\); Sketch a possible graph of \(f(x)\) and justify your answer.

Step-by-Step Solution

Answer

The graph of \( f(x) \) is symmetric around the y-axis and has a minimum at \( x = 0 \) with increasing behavior away from \( x = 0 \).

Step 1: Understanding Function g(x)

The function \( g(x) \) is given with specific properties: it is concave down when \( x < 1 \), concave up when \( x > 1 \), has a positive derivative everywhere except at \( x = 1 \), and \( g(1) = 1 \). This suggests \( g(x) \) has a minimum at \( x = 1 \) and is likely an even function due to the symmetry in concavity around \( x = 1 \).

Step 2: Evaluate Function f(x)

The function \( f(x) = g(x^4) \). We need to understand that \( x^4 \) is non-negative for all \( x \). Hence, \( g(x^4) \) will reflect the behavior of \( g(x) \) at positive values for all negative and positive \( x \). Additionally, the symmetry of \( x^4 \) (since \( x^4 = (-x)^4 \)) means \( f(x) \) is symmetric around the y-axis.

Step 3: Characteristics of f(x) Derived from g(x)

Because \( g(x^4) \) simplifies to \( g(y) \) where \( y = x^4 \), \( f(x) = g(y) \) satisfies similar properties. Since \( g \) is increasing for all \( x \), \( f(x) \) is also increasing except at \( x = 0 \), where \( x^4 = 0 \). The point at \( x = 0 \, ( \text{since } x^4 = 0 \text{ only at } x = 0 ) \) is analogous to \( x = 1 \) in \( g \). Thus, \( f \) has a minimum at \( x = 0 \).

Step 4: Graph Sketch Justification

The properties of \( g(x) \) translate through \( f(x) = g(x^4) \), resulting in \( f(x) \) being symmetric about the y-axis, with a global minimum at \( x = 0 \) and being increasing for \( x eq 0 \). At \( x = 1 \text{ or } -1 \), \( f(x) = g(1) = 1 \). Since \( g \) is concave up for \( y > 1 \) (i.e., \( x^4 > 1 \)) and concave down for \( y < 1 \) (i.e., \( 0 < x^4 < 1 \)), the concavity affects \( f(x) \) similarly. Hence, the graph of \( f \) will have its lowest point at the origin, becoming progressively steeper as \( |x| \) increases away from zero.

Key Concepts

Understanding Derivative Properties

Derivatives are fundamental in analyzing the behavior of functions, telling us about both their slope and concavity. In the exercise given, it is crucial to observe that the derivative of the function \( g(x) \) is positive everywhere except at \( x = 1 \).
This means that \( g(x) \) is increasing at all points except possibly at this single point. With these characteristics, a derivative helps understand where functions rise, fall, or stay constant. Generally:

• If \( g'(x) > 0 \), the function is increasing at that interval.
• If \( g'(x) = 0 \), there could be a halt in the increase or decrease, marking a possible point of local extremum.
• If \( g'(x) < 0 \), the function is decreasing at that interval.

Through this lens, knowing the derivative signs gives insights into the graph's shape and the behavior of the function as it moves across the x-axis. This positive derivative almost everywhere, except at one point, implies that \( g(x) \), and subsequently \( f(x) \), are largely increasing.

Exploring Symmetry in Functions

Symmetry plays a vital role in understanding how functions behave across their domains. It simplifies graph analysis as it often means that behaviors on one side reflect similarly on the other side.
In the context of the given problem, due to the symmetry of \( g(x) \) implied by its concavity properties and \( x^4 \)'s inherent even nature, the resulting function \( f(x) = g(x^4) \) is symmetric around the y-axis.
Some characteristics of symmetry include:

• If a function is even, \( f(x) = f(-x) \), it means that the left side of the graph mirrors the right side.
• If a function is odd, \( f(-x) = -f(x) \), the graph is symmetrical about the origin.

For \( f(x) \), the symmetry ensures that any feature on the positive side of y-axis is mirrored on the negative side. This reduces the need to analyze both sides separately, as behavior is consistent around the y-axis.

Concave Function Characteristics Simplified

Concavity gives us critical insights about the function's behavior regarding how its slope changes over an interval. Functions can be concave up or concave down. In this exercise, the function \( g(x) \) splits this concavity mid-way:

• Concave Down: for \( x < 1 \) in \( g(x) \)—like an upside-down bowl, indicating the slope decreases in this region.
• Concave Up: for \( x > 1 \) in \( g(x) \)—like a regular bowl, indicating the slope is increasing in this region.

Here's how it looks practically:- **Concave Down**: Think of a mountain peak at x = 1. It's the highest point before the slope starts decreasing.- **Concave Up**: Once past x = 1, the function shape begins to look like the bottom of a hill, increasing slowly to quickly.
Understanding these transitions helps in gauging the steepness and shape of \( f(x) \) where the inputs are transformed through \( x^4 \). Therefore, across positive \( x^4 \) values, \( f(x) \) will reflect the behavior of \( g(x) \) based on these concave shifting properties, resulting in an intuitive demarcation of curvature in its graph.