Problem 247 For the following exercises, con... [FREE SOLUTION]

Chapter 4: Problem 247

For the following exercises, consider a third-degree polynomial $f(x), \quad$ which has the properties $f^{\prime}(1)=0, f^{\prime}(3)=0$ . Determine whether the following statements are true or false. Justify your answer. $$ f^{\prime \prime}(x)=0 \text { for some } 1 \leq x \leq 3 $$

Short Answer

Expert verified

True: $f''(x) = 0$ at $x = 2$ within $1 \leq x \leq 3$.

Step by step solution

Understand the Problem

We are given a third-degree polynomial $f(x)$ with the derivative root at $x = 1$ and $x = 3$. We need to determine if there exists some $x$ in the interval $1 \leq x \leq 3$ where the second derivative $f''(x) = 0$.

Application of Rolle's Theorem

Rolle's Theorem states that if a function is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there is at least one $c$ in $(a,b)$ such that $f'(c) = 0$. Since $f'(1) = 0$ and $f'(3) = 0$, Rolle's Theorem suggests there must be a $c$ in $1 < c < 3$ where $f''(c) = 0$.

Analyzing the Derivative

Since $f(x)$ is a third-degree polynomial, $f'(x)$ is a second-degree polynomial. The solutions $x = 1$ and $x = 3$ suggest $f'(x) = k(x - 1)(x - 3)$ for some constant $k$ (i.e., $f'(x)$ is a quadratic polynomial with roots at $1$ and $3$).

Deriving the Second Derivative

Differentiate $f'(x) = k(x - 1)(x - 3)$ with respect to $x$ to get the second derivative $f''(x) = k(2x - 4)$.

Solving for When $f''(x) = 0$

Set $f''(x) = k(2x - 4) = 0$ and solve for $x$:\[2x - 4 = 0 \x = 2\]

Verify $x = 2$ is in the Interval

Since $x = 2$ lies within $1 \leq x \leq 3$, the second derivative indeed equals zero for some $x$ in the interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Third-Degree Polynomial

A third-degree polynomial, often referred to as a cubic polynomial, is an algebraic expression of the form:\[ f(x) = ax^3 + bx^2 + cx + d \]where $ a, b, c, $ and $ d $ are constants and $ a eq 0 $. This polynomial is characterized by its cubic term, $ ax^3 $, which gives it a distinctive curve with up to three roots and two turning points, often forming an \'S\'-shaped curve.

Key Features:

It has three roots, which are the x-values where the polynomial equals zero.
It can have up to two turning points, which are peaks or valleys in the curve.
The leading coefficient $ a $ affects the direction and width of the curve.

Understanding the properties of a third-degree polynomial is crucial when analyzing its derivatives, as these derivatives provide insight into the nature of the function's graph.

Second Derivative

The second derivative of a function like a third-degree polynomial provides information about the curvature or concavity of the graph. For a function $ f(x) $, the second derivative is denoted $ f''(x) $. It is the derivative of the first derivative, essentially describing how the slope of the curve changes.

Importance:

A positive second derivative $ f''(x) > 0 $ indicates that the graph is concave up, or like a cup.
A negative second derivative $ f''(x) < 0 $ indicates concave down, or like a cap.
When $ f''(x) = 0 $, it could signal an inflection point, where the concavity changes.

In the context of the given problem, finding where $ f''(x) = 0 $ suggests that there might be an inflection point within the interval $[1, 3]$, as determined using Rolle's Theorem.

Quadratic Polynomial

A quadratic polynomial is a second-degree polynomial, which typically takes the form:\[ g(x) = ax^2 + bx + c \]where $ a, b, $ and $ c $ are constants and $ a eq 0 $. In the context of the exercise, when we derived the first derivative of the third-degree polynomial $ f(x) $, it resulted in a quadratic polynomial.

Characteristics:

It has up to two roots and one vertex, which is either a minimum or maximum point.
The coefficient $ a $ determines whether the parabola opens upward ($ a > 0 $) or downward ($ a < 0 $).
The graph of a quadratic polynomial is a parabola.

In relation to the problem, since $ f'(x) $ is quadratic with roots at $ x = 1 $ and $ x = 3 $, it implies that the change in slope across these roots leads to identifying the x value where $ f''(x) = 0 $. This insight is vital for understanding where the concavity alters within the specified interval.

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91影视

For the following exercises, consider a third-degree polynomial \(f(x), \quad\) which has the properties \(f^{\prime}(1)=0, f^{\prime}(3)=0\) . Determine whether the following statements are true or false. Justify your answer. $$ f^{\prime \prime}(x)=0 \text { for some } 1 \leq x \leq 3 $$

Short Answer

Step by step solution

Understand the Problem

Application of Rolle's Theorem

Analyzing the Derivative

Deriving the Second Derivative

Solving for When \(f''(x) = 0\)

Verify \(x = 2\) is in the Interval

Key Concepts

Third-Degree Polynomial

Second Derivative

Quadratic Polynomial

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