Problem 46 - 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 of \(f\) equals its quadratic approximation because \(f''(a) = 0\), removing the quadratic term.

Step 1: Define key terms

An inflection point is where the concavity of the function changes, which implies that the second derivative is zero at that point. Therefore, for a function \(f(x)\) with a twice-differentiable function, if \(a\) is an inflection point, \(f''(a) = 0\). The linearization of \(f(x)\) at \(x = a\) is given by: \[ L(x) = f(a) + f'(a)(x - a) \].

Step 2: Quadratic approximation of function

The quadratic approximation of a function \(f(x)\) around \(x = a\) is given by: \[ Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x-a)^2 \]. At an inflection point \(x = a\), we know \(f''(a) = 0\), so the quadratic approximation simplifies to \[ Q(x) = f(a) + f'(a)(x - a) \].

Step 3: Compare Linearization and Quadratic Approximation

From the expressions, the linearization \(L(x)\) is: \[ L(x) = f(a) + f'(a)(x - a) \]. Comparing it with the quadratic approximation \(Q(x) = f(a) + f'(a)(x - a)\), both are identical. Hence, at an inflection point, the linearization and quadratic approximation of the function are the same.

Key Concepts

Linearization

Linearization is a powerful tool in calculus that helps to approximate a complex function with a simpler one, typically a straight line. This is particularly useful when we want to estimate the value of a function near a specific point. The linearization of a function at a certain point is essentially the equation of the tangent line to the function at that point.

When we take the linearization of a function, we're using the first derivative to capture the slope of the function at the exact point of interest. The formula for linearization at point \( x = a \) is given by:

• \( L(x) = f(a) + f'(a)(x - a) \)

Here, \( f(a) \) is the function's value at \( x = a \), and \( f'(a) \) is the slope of the tangent line at that point.

Linearization is especially effective at inflection points. At an inflection point, the function changes its concavity, and surprisingly, the tangent perfectly approximates the curve, matching both the linear and quadratic perspectives.

Quadratic Approximation

Quadratic approximation extends the idea of linearization by including the curvature of the function. While linearization only accounts for the slope, quadratic approximation considers both the slope and the concavity at a given point. Its formula around a point \( x = a \) is:

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

The quadratic term \( \frac{f''(a)}{2}(x-a)^2 \) reflects how the function curves around \( a \).

At inflection points, the second derivative \( f''(a) \) equals zero, eliminating the quadratic term from the formula. This means:

• \( Q(x) = f(a) + f'(a)(x - a) \)

Therefore, at inflection points, the linearization and quadratic approximation are the same, making the tangent line an excellent fit for predicting the behavior of the function around that point.

Concavity

Concavity describes the direction and rate of change of a curve. When we analyze a function, we often look at its concavity to understand its shape and how it behaves as it varies in the domain. Concavity is determined by the second derivative \( f''(x) \).

• If \( f''(x) > 0 \), the function is concave up, resembling a U-shape.
• If \( f''(x) < 0 \), the function is concave down, akin to a n-shape.
• When \( f''(x) = 0 \), the function's concavity changes, indicating a possible inflection point.

An inflection point occurs where \( f''(x) \) changes sign, pointing to a shift from concave up to concave down or vice versa. At these points, interestingly, the graph of the function seems to align perfectly with its tangent, as both the linearization and quadratic approximation coincide, enhancing our understanding of these crucial boundary moments in a function's graph.