Problem 21 - University Calculus: Early Transcendentals

Question

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x^{5}-5 x^{4}=x^{4}(x-5)$$

Step-by-Step Solution

Answer

Local min at (4, -256). Inflection points at (0,0) and (3,-162).

Step 1: Find Critical Points

\(y' = 5x^4 - 20x^3 = 5x^3(x-4)\)
\(y' = 0\) at \(x = 0\) and \(x = 4\).

Step 2: Determine Extreme Values

Sign of \(y'\): negative for \(x < 0\), negative for \(0 < x < 4\), positive for \(x > 4\).
At \(x = 0\): no sign change, so no local extremum (inflection point).
At \(x = 4\): changes from \(-\) to \(+\), local minimum. \(y(4) = 1024 - 1280 = -256\).

Step 3: Find Inflection Points

\(y'' = 20x^3 - 60x^2 = 20x^2(x-3)\)
\(y'' = 0\) at \(x = 0\) and \(x = 3\).
At \(x = 3\): \(y'' \) changes sign, so inflection point. \(y(3) = 243 - 405 = -162\).
At \(x = 0\): \(y''\) does not change sign (stays \(\leq 0\)), so \(x = 0\) is also an inflection point.

Step 4: Summary

Local minimum at \((4, -256)\). Inflection points at \((0, 0)\) and \((3, -162)\). No absolute extremes.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes at any given point. Think of a derivative as the "slope" or "rate of change" of a curve.
For a polynomial function, the derivative is calculated by applying power rules where each term of the expression is differentiated separately.
This is crucial in finding extreme points, as they occur where the derivative equals zero. In our case, the function is given as \( y = x^5 - 5x^4 = x^4(x - 5) \). The first derivative, found through differentiation, is \( y' = 5x^4 - 20x^3 \).
This derivative helps us identify where the function's rate of change is zero, which are potential points for finding local maxima and minima.

Extreme Points

Extreme points are where a function reaches its highest or lowest value in a localized region, known as local maxima and minima. Finding these points involves setting the first derivative equal to zero.
For our function, we already found the derivative to be \( y' = 5x^4 - 20x^3 \). To find extreme points, solve \( 5x^4 - 20x^3 = 0 \). Factoring gives \( 5x^3(x - 4) = 0 \).
This indicates possible extreme points at \( x = 0 \) and \( x = 4 \). We then use the second derivative test or analyze the behavior around these points to determine if they are maxima, minima, or saddle points. Points where the function changes from increasing to decreasing are maxima, and where it changes from decreasing to increasing are minima.

Inflection Points

An inflection point is where the function changes its concavity. That is, where it shifts from being concave up (like a cup) to concave down (like a frown), or vice versa. To find inflection points, we calculate the second derivative and set it to zero. For our function, after calculating the first derivative as \( y' = 5x^4 - 20x^3 \), find the second derivative by differentiating this derivative.
The second derivative will help identify where the concavity changes. Solve this second derivative to find the inflection points if they exist. Examining the signs before and after these points clarifies the type of concavity transition, marking the inflection points.
This provides deeper insights into the behavior of the polynomial's graph, offering a detailed perspective on where the function's curvature changes.

Polynomials

Polynomials are algebraic expressions made up of variables and coefficients, constructed using operations like addition, subtraction, multiplication, and exponentiation with non-negative integer exponents.
The function \( y = x^5 - 5x^4 \) is a polynomial of degree 5, meaning the highest power of \( x \) is 5.
The degree affects the number of roots and shape of the graph. Higher-degree polynomials can have complex shapes with multiple turning points and inflection points.

• The leading term indicates the end behavior of the polynomial, determining how the graph behaves as \( x \) goes to infinity or negative infinity.
• The number of solutions, including possible repeated solutions, hinges on the Fundamental Theorem of Algebra which states a polynomial of degree \( n \) has exactly \( n \) complex roots.

Experimenting and exploring polynomials reveal intriguing patterns and sutler behaviors, making them a cornerstone of calculus and algebra studies.