Problem 54 - Calculus: Early Transcendentals

Question

Describe the concavity of \(y=x^{3}+b x^{2}+c x+d\). You will need to consider different cases, depending on the values of the coefficients.

Step-by-Step Solution

Answer

The concavity depends on \( x > -\frac{b}{3} \) for concave up and \( x < -\frac{b}{3} \) for concave down.

Step 1: Find the Second Derivative

To determine the concavity of the function, we first need its second derivative. The original function is given as \( y = x^3 + bx^2 + cx + d \). First, find the first derivative:\[ y' = 3x^2 + 2bx + c \]Then, find the second derivative:\[ y'' = 6x + 2b \]

Step 2: Determine Concavity

The concavity of the function depends on the sign of the second derivative, \( y'' = 6x + 2b \).- The function is **concave up** where \( y'' > 0 \): \[ 6x + 2b > 0 \Rightarrow x > -\frac{b}{3} \]- The function is **concave down** where \( y'' < 0 \): \[ 6x + 2b < 0 \Rightarrow x < -\frac{b}{3} \]

Step 3: Identify the Point of Inflection

The point where the concavity changes is the point of inflection. This occurs when the second derivative is zero:\[ 6x + 2b = 0 \Rightarrow x = -\frac{b}{3} \]At this value of \( x \), the concavity of the function changes.

Step 4: Analyze Special Cases

Consider special cases when \( b = 0 \): - If \( b = 0 \), the second derivative simplifies to \( y'' = 6x \), making the concavity dependent solely on \( x \). In this case, the function is concave up for \( x > 0 \) and concave down for \( x < 0 \), with the point of inflection at \( x = 0 \).

Key Concepts

Second Derivative

The second derivative of a function is essential when analyzing concavity. To find the concavity of a polynomial function like \( y = x^3 + bx^2 + cx + d \), the second derivative, denoted as \( y'' \), offers insight. By calculating it, we can determine intervals of concavity:

• Concave Up: This happens on intervals where \( y'' > 0 \). For our example, \( y'' = 6x + 2b \), meaning we check \( 6x + 2b > 0 \).
• Concave Down: Occurs on intervals where \( y'' < 0 \), which translates to \( 6x + 2b < 0 \).

Calculating these inequalities gives you the ranges where the graph opens upwards or downwards, indicating the function's behavior.

Point of Inflection

The point of inflection is a fascinating concept where a function changes its concavity, meaning it goes from concave up to concave down or vice versa. To find this point, solve for when the second derivative equals zero.For the polynomial \( y = x^3 + bx^2 + cx + d \), solve \( 6x + 2b = 0 \). This gives us the x-value of the inflection point, \( x = -\frac{b}{3} \). At this point:

• The concavity shifts.
• The nature of the curve changes direction in terms of upwards or downwards bending.

Recognizing the point of inflection helps anticipate changes in the function's growth pattern.

Polynomial Function

Polynomial functions like \( y = x^3 + bx^2 + cx + d \) are comprised of variables raised to any power of the natural numbers. These functions are fundamental in calculus for a multitude of reasons:

• Structure: Consist of terms where variables are raised to powers, with coefficients that can vary.
• Behavior: Their differentiability offers insights via derivatives, making them ideal for studying changes in slopes and concavity.

The polynomial function in our exercise is a cubic polynomial, with a degree of 3. This means it is one of the simplest forms able to demonstrate points of inflection. Understanding its shape and roots involves analyzing the role each coefficient plays in shaping the function's graph.