Problem 113 - Thomas Calculus

Question

If \(b, c,\) and \(d\) are constants, for what value of \(b\) will the curve \(y=x^{3}+b x^{2}+c x+d\) have a point of inflection at \(x=1 ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

The value of \(b\) is \(-3\).

Step 1: Review Inflection Point Condition

A point of inflection occurs where the concavity of the curve changes, which happens when the second derivative of the function changes sign. This means that at a point of inflection, the second derivative is equal to zero.

Step 2: Find the First Derivative

The first derivative of the given curve is found to understand the slope change rate. Given the curve, find its first derivative:\[ y = x^3 + bx^2 + cx + d \]Taking the derivative:\[ y' = 3x^2 + 2bx + c \]

Step 3: Find the Second Derivative

Now, differentiate the first derivative to find the second derivative:\[ y' = 3x^2 + 2bx + c \]Differentiating again, we obtain:\[ y'' = 6x + 2b \]

Step 4: Apply Inflection Condition

Apply the inflection condition which states that the second derivative should be zero at the point of inflection. Thus, set the second derivative equal to zero at \(x = 1\):\[ 6(1) + 2b = 0 \]Simplifying, we have:\[ 6 + 2b = 0 \]

Step 5: Solve for b

Solve the equation from Step 4 for the value of \(b\):\[ 6 + 2b = 0 \]\[ 2b = -6 \]\[ b = -3 \]

Key Concepts

Second Derivative

The second derivative of a function is a crucial concept in calculus, especially when analyzing the behavior of curves. The second derivative, often denoted as \( y'' \), provides insight into the "acceleration" of a curve or how the slope of the tangent line itself is changing.

Here’s what to remember about the second derivative:

• Definition: It is the derivative of the first derivative of a function. In essence, you take the derivative of the function twice.
• Usage: It helps us determine the concavity (whether the curve bends upwards or downwards) and identify inflection points where concavity changes.
• Expression: For the cubic function \(y = x^3 + bx^2 + cx + d\), its second derivative is \(y'' = 6x + 2b\).

Understanding the second derivative is important to effectively find points of inflection, which occur when the second derivative is zero and the sign of the second derivative changes.

Concavity

Concavity describes how a curve bends or "opens". A curve can be either concave up or concave down, depending on the behavior of its second derivative.

Here’s how concavity works:

• Concave Up: When the second derivative \(y'' > 0\), the curve is concave up, resembling a smile (>).
• Concave Down: When \(y'' < 0\), the curve is concave down, resembling a frown (∩).
• Inflection Points: Occur where the curve changes concavity. This means the second derivative must equal zero, and you should check that the sign of \(y''\) actually changes around this point.

For the function in our example, the curve changes concavity at \(x = 1\), making it a candidate for an inflection point. At this point, the second derivative equation \(6x + 2b = 0\) is zero, confirming it is an inflection point when \(b = -3\).

Cubic Function

A cubic function is a polynomial function of degree three, and it generally has the form \(y = ax^3 + bx^2 + cx + d\). These functions have distinct characteristics that influence their graphs and points of inflection.

Key features of cubic functions include:

• Degree Three: Being a third-degree polynomial, they can have up to three real roots and can cross the \(x\)-axis up to three times.
• S-shape: Graphs of cubic functions typically have an "S" shape, guiding us through a fan of curves bent by concave up and concave down sections.
• Inflection Points: These points, such as \(x = 1\) in our exercise, are very important because they indicate where the S-shape changes direction.

For our particular cubic function \(y = x^3 + bx^2 + cx + d\), determining \(b\) such that \(x = 1\) is an inflection point is key to adjusting its graphical properties correctly.