Problem 223 - Calculus Volume 1

Question

Determine a. intervals where \(f\) is concave up or concave down, and b. the inflection points of \(f\). \(f(x)=x^{3}-4 x^{2}+x+2\)

Step-by-Step Solution

Answer

Concave down on \((-\infty, \frac{4}{3})\), concave up on \((\frac{4}{3}, \infty)\); inflection point at \(\left(\frac{4}{3}, -\frac{10}{27}\right)\).

Step 1: Find the Second Derivative

To determine where the function is concave up or concave down, and to find the inflection points, we need to find the second derivative of the function. Start by computing the first derivative:\[ f'(x) = 3x^2 - 8x + 1 \]Then, find the second derivative:\[ f''(x) = 6x - 8 \]

Step 2: Set the Second Derivative Equal to Zero

Inflection points occur where the second derivative changes sign, i.e., the points where \( f''(x) = 0 \). Set the second derivative to zero and solve for \( x \):\[ 6x - 8 = 0 \]Solving for \( x \), we get:\[ 6x = 8 \]\[ x = \frac{4}{3} \]

Step 3: Test Intervals for Concavity

Using the result from Step 2, test the intervals around \( x = \frac{4}{3} \) to determine where the function is concave up or concave down. Choose test points in the intervals created by the potential inflection point:1. Interval \( (-\infty, \frac{4}{3}) \): Choose \( x = 0 \), then \( f''(0) = 6(0) - 8 = -8 \), indicating that the function is concave down on \( (-\infty, \frac{4}{3}) \).2. Interval \( (\frac{4}{3}, \infty) \): Choose \( x = 2 \), then \( f''(2) = 6(2) - 8 = 4 \), indicating that the function is concave up on \( (\frac{4}{3}, \infty) \).

Step 4: Identify the Inflection Points

An inflection point occurs at \( x = \frac{4}{3} \) because the second derivative changes sign at this point. To find the exact coordinates of the inflection point, evaluate the original function at \( x = \frac{4}{3} \):\[ f\left(\frac{4}{3}\right) = \left(\frac{4}{3}\right)^3 - 4\left(\frac{4}{3}\right)^2 + \frac{4}{3} + 2 \]Simplify:\[ = \frac{64}{27} - \frac{64}{9} + \frac{4}{3} + 2 \approx -\frac{10}{27} \]Thus, the inflection point is \( \left(\frac{4}{3}, -\frac{10}{27}\right) \).

Key Concepts

Concavity

Concavity describes how a curve bends. It reveals if a function is "bowed" upwards or downwards.
To understand concavity, you look at the second derivative of a function.

• If the second derivative is positive over an interval, the function is concave up on that interval. This means the graph "opens" upwards like a smile.
• If the second derivative is negative, the function is concave down. The graph "opens" downwards like a frown.

This helps in understanding the shape of a graph and its potential turning points.

Inflection Points

Inflection points are where the curve changes concavity. Think of it as a switch where the graph goes from curving up to curving down, or vice versa. To find inflection points:

• Identify where the second derivative equals zero (\(f''(x) = 0\)).
• Ensure there is a change in the concavity (sign of the second derivative) around these points.

Finding inflection points provides insights on where the function's direction or "bend" changes.

Second Derivative

The second derivative, noted as \(f''(x)\), represents the rate of change of the first derivative. It essentially provides the acceleration of the function's slope.

• Calculating the second derivative helps in analyzing concavity and identifying inflection points.
• For the given polynomial \(f(x)=x^{3}-4x^{2}+x+2\), the second derivative \(f''(x)=6x-8\), helps detect where the graph changes its concavity.

The second derivative is a vital tool for understanding the "curviness" of graphs.

Polynomial Functions

Polynomial functions, like \(f(x)=x^{3}-4x^{2}+x+2\), are expressions with multiple terms, consisting of variables and coefficients. Each variable is raised to a whole number power.

• These functions can have multiple terms and varying degrees of highest powers, allowing them to form different shapes and flexibilities on the graph.
• Calculus offers powerful techniques to analyze polynomial behaviors, such as finding derivatives to understand concavity and inflection points.

Polynomials are fundamental in modeling real-world phenomena and offer insights into smooth changes and behaviors of different functions on their intervals.