Problem 54 - Calculus an Applied Approach

Question

determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(x+1)(x+2)(x+3)(x+4), \text { then } \frac{d^{5} y}{d x^{5}}=0 $$

Step-by-Step Solution

Answer

The statement is true. The fifth derivative of the function \(y = x^4 + 10x^3 + 35x^2 + 50x + 24\) is indeed zero.

Step 1: Derive the polynomial function

We start by expanding the given function \(y=(x+1)(x+2)(x+3)(x+4)\) and simplifying it. The simplified form of the polynomial should be \(y = x^4 + 10x^3 + 35x^2 + 50x + 24\)

Step 2: Compute the fifth derivative

Now compute the fifth derivative of the function. Given the degree of the polynomial is 4, from differentiation property we understand that the derivative of a polynomial of degree 'n' will result in a polynomial of degree 'n-1'. This means that the fifth derivative of our fourth degree polynomial will yield constant, which will then be zero when differentiated further.

Step 3: Evaluate the statement

The fifth derivative of the function \(y = x^4 + 10x^3 + 35x^2 + 50x + 24\) is zero, hence the given statement is true.

Key Concepts

Fifth Derivative

When we talk about taking the derivative of a function, we're essentially finding out how the function changes. The derivative provides us with the rate at which the function value changes as its input changes. Now, imagine doing this repeatedly. The fifth derivative simply means we are finding the derivative of a derivative, five times in a row.

If you have a polynomial function like \( y = (x+1)(x+2)(x+3)(x+4) \), after expanding and simplifying, we get a fourth-degree polynomial \( y = x^4 + 10x^3 + 35x^2 + 50x + 24 \).

• First derivative: reduces degree by one, giving us a third-degree polynomial.
• Second derivative: further reduces the degree, resulting in a second-degree polynomial.
• Continuing this, the fourth derivative results in a constant, because it reduces a first-degree polynomial.
• Thus, the fifth derivative of this particular feature becomes zero.

Here, the trick is to remember that repeatedly differentiating boils down to decreasing the degree of the polynomial until it disappears, if you start lower than the derivative count.

Degree of Polynomial

The degree of a polynomial is essentially the highest power of the variable within the polynomial expression. It tells us a lot about the function's behavior and its derivatives. For our function, which simplifies into \( y = x^4 + 10x^3 + 35x^2 + 50x + 24 \), the highest power of \( x \) is 4. Therefore, we say it is a fourth-degree polynomial.

Understanding the degree:

• A higher degree polynomial's graph can have more curves (number of real roots lines).
• The degree also guides how often you can differentiate before reaching zero.
• Therefore, since our polynomial is degree 4, the fifth derivative will be zero as each derivative reduces the degree by one: 4 to 3, 3 to 2, 2 to 1, 1 to 0, and finally zero.

Knowing this, we save time and effort in deriving continuously till it becomes overly complex.

Differentiation Property

Differentiation involves applying certain rules to systematically transform a function. One important property to remember is that the derivative of a polynomial of degree \( n \) is another polynomial of degree \( n - 1 \).

This property is immensely helpful when predicting outcomes of derivatives without completely calculating them. Here's a clear picture:

• For a polynomial \( ax^n + bx^{n-1} + \,\ldots\, + c \), the first derivative will be \( n*ax^{n-1} + (n-1)*bx^{n-2} + \,\ldots\, \).
• Continuing differentiation ultimately leads the polynomial degree to diminish till it hits zero, ending up in constant zero.

Digesting this fundamental differentiation property simplifies tasks like quickly determining if higher order derivatives, like the fifth derivative of our polynomial, will result in zero.

True or False Statements

When working with polynomial differentiation, we often encounter statements that require proof of being true or false. A classic example is the statement about the fifth derivative becoming zero for polynomials of lower degree than five.

To evaluate such statements, it is effective to break down the problem and use mathematical properties such as degree and differentiation rules:

• The problem asked us if the fifth derivative of \( y=(x+1)(x+2)(x+3)(x+4) \) is zero.
• Once the polynomial is expanded and we identify it as a fourth-degree polynomial, we understand that its fifth derivative will blank out into zero.
• Thus, the statement was verified to be true through understanding of polynomial properties.

Using these breakdowns and systematic analysis, we learn to confidently assert the truthfulness of mathematical statements.