Problem 53 - Calculus

Question

Decide if the statements in Problems are true or false. Give an explanation for your answer. The quadratic approximation to \(f(x)\) near \(x=-1\) shows that if \(f(-1)=g(-1), f^{\prime}(-1)=g^{\prime}(-1),\) and \(f^{\prime \prime}(-1) < g^{\prime \prime}(-1),\) then \(f(x) < g(x)\) for all \(x\) sufficiently close to -1 (but not equal to -1 ).

Step-by-Step Solution

Answer

The statement is true, as evidenced by the smaller second derivative of \( f(x) \) slowing its growth near \( x = -1 \).

Step 1: Understand Quadratic Approximation

Quadratic approximation uses the second-degree Taylor polynomial to estimate the behavior of a function near a point. For a function \( f(x) \), its quadratic approximation near \( x = a \) is \( f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2 \). The same applies for \( g(x) \).

Step 2: Compare Functions at x = -1

Given \( f(-1) = g(-1) \) and \( f'(-1) = g'(-1) \), the functions are the same for constant and linear terms in their Taylor expansion at \( x = -1 \), meaning their values and first derivatives are identical at \( x = -1 \).

Step 3: Analyze the Second Derivative Influence

Since \( f''(-1) < g''(-1) \), the quadratic (second degree) term in the expansion for \( f(x) \) is smaller than that for \( g(x) \) near \( x = -1 \). The quadratic term contributes differently to \( f(x) \) and \( g(x) \), modifying the steepness around \( x = -1 \).

Step 4: Determine the Effects Near x = -1

The inequality \( f''(-1) < g''(-1) \) implies that for \( x \) near -1 (but not equal to -1), \( f(x) \) will grow slower compared to \( g(x) \) due to its smaller second derivative, leading \( f(x) < g(x) \).

Step 5: Conclusion on the Statement

Since \( f''(-1) < g''(-1) \) and all other terms are initially equal, \( f(x) < g(x) \) holds true for \( x \) sufficiently close to \( -1 \). Thus, the given statement is true.

Key Concepts

Taylor Polynomial

When studying how functions behave near a certain point, a powerful tool in calculus is the Taylor polynomial. Specifically, the second-degree Taylor polynomial, also known as the quadratic approximation, helps approximate complex functions with simpler, quadratic forms. This provides a more manageable expression when analyzing functions around a particular point, say \( x = a \). The formulation involves taking into account the value of the function itself, the first derivative (related to the slope), and the second derivative (related to curvature).

• The quadratic approximation is given by: \( f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2 \).
• Each term contributes to how well the polynomial matches the original function; especially as we consider points closer and closer to \( a \).

The idea is not just to provide an exact copy of the function but rather a simplified expression that behaves similarly close to the point \( a \), capturing vital elements of the function's shape.

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), is a crucial part of understanding its behavior near a certain value. In the context of quadratic approximations, it tells us about the concavity of the function. How curved the function appears can dictate how it compares to other functions.

• If \( f'' > 0 \), the function is concave up, resembling a U-shape.
• If \( f'' < 0 \), the function is concave down, resembling an inverted U-shape.

In the given exercise, where \( f''(-1) < g''(-1) \), it shows that "\( f(x)\)" grows slower than "\( g(x)\)" around \( x = -1\). This smaller second derivative at \( x = -1 \) makes a significant difference in how much the function "bends" and influences the relationship between the functions in question.

Function Behavior Analysis

Analyzing how functions behave near specific points is essential in mathematical studies, especially when we want to compare them precisely. This involves looking at initial values, how they initially move or tilt, and how they bend.

• Start by ensuring the functions have the same initial point and initial slope, as seen with equal value \( f(-1) = g(-1) \) and first derivative \( f'(-1) = g'(-1) \).
• Next, observe the second derivative, since any difference here affects how much the functions diverge.

In our context, although both functions start identically at \( x = -1 \), the inequality in second derivatives \( f''(-1) < g''(-1) \) suggests they will not stay equal as you move slightly away. This is because the function with the lower second derivative, \( f(x) \), does not curve as much, thus staying below \( g(x) \) around the point \( x = -1 \), leading to \( f(x) < g(x) \). This simple yet powerful analysis allows us to draw conclusions about function behavior from just their derivatives.