Problem 50 - Thomas Calculus

Question

Linearizations at inflection points Show that if the graph of a twice- differentiable function \(f(x)\) has an inflection point at \(x=a\) , then the linearization of \(f\) at \(x=a\) is also the quadratic approximation of \(f\) at \(x=a\) . This explains why tangent lines fit so well at inflection points.

Step-by-Step Solution

Answer

At an inflection point, the linearization and quadratic approximation equal because the second derivative term is zero.

Step 1: Understand inflection points

An inflection point is where the graph of a function changes concavity, meaning that the second derivative changes sign. If the graph has an inflection point at \(x=a\), then \(f''(a) = 0\).

Step 2: Define linearization

The linearization of a function \(f\) at \(x=a\) is given by the formula \(L(x) = f(a) + f'(a)(x-a)\). This is essentially the equation of the tangent line at that point.

Step 3: Define quadratic approximation

The quadratic approximation of function \(f\) at \(x=a\) uses the Taylor series expansion up to the second derivative: \(Q(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\).

Step 4: Use the condition at inflection point

At the inflection point, \(f''(a) = 0\). Substituting this into the quadratic formula, we have \(Q(x) = f(a) + f'(a)(x-a) + 0 \cdot (x-a)^2 = f(a) + f'(a)(x-a)\).

Step 5: Compare linearization and quadratic approximation

Both the linearization \(L(x) = f(a) + f'(a)(x-a)\) and the quadratic approximation \(Q(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\) simplify to the same expression at the inflection point because the \(\frac{f''(a)}{2}(x-a)^2\) term vanishes.

Key Concepts

Understanding Linearization

Linearization is a way to approximate a function using a linear model. It simplifies a complex function around a specific point, making calculations easier. To linearize a function, we use the tangent line approximation.
This involves the first derivative of the function, creating a straight line that locally resembles the function.
The formula for linearization at a point \(x = a\) is given by:

• \(L(x) = f(a) + f'(a)(x-a)\)

This equation reflects the value of the function at \(a\), plus the slope \(f'(a)\), which represents how steep the function is at point \(a\).
Effectively, linearization helps in understanding how a function behaves near a specific point without dealing with its overall complexity.

Exploring Quadratic Approximation

Quadratic approximation is a step beyond linearization. It offers a more accurate estimation by considering not just the first derivative, but also the second derivative of a function.
This approximation is part of a Taylor series expansion and is very useful near the point of interest.
For a function \(f(x)\) approximated at \(x = a\), the quadratic approximation formula is:

• \(Q(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2\)

Here, \(f''(a)\) indicates the concavity of the function. At an inflection point, this second derivative is zero, which simplifies the approximation to a linear one.
Thus, at inflection points, quadratic and linear approximations are the same because the term with \(f''(a)\) vanishes.

Importance of Concavity

Concavity provides insight into the shape of a graph. It tells us whether the graph curves upwards or downwards at a specific point. The second derivative \(f''(x)\) helps determine this behavior.
- If \(f''(x) > 0\), the function is concave upward, resembling a "U" shape.- If \(f''(x) < 0\), it's concave downward, like an "n" shape.An inflection point is where the function changes from concave up to concave down, or vice versa.
Here, the second derivative equals zero, \(f''(a) = 0\). This is crucial for the concept discussed, as inflection points simplify quadratic approximations, showing why linearization fits perfectly at these points.
Understanding concavity helps in graph interpretation and is essential in optimization and calculus applications.